LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


Class 


s$a^ 


HYDRAULICS 

OF 

RIVERS,  WEIRS  AND  SLUICES 


THE    DERIVATION    OF   NEW   AND   MORE  ACCURATE 

FORMULAE,  FOR  DISCHARGE  THROUGH  RIVERS 

AND    CANALS    OBSTRUCTED    BY    WEIRS, 

SLUICES,  ETC.,  ACCORDING  TO  THE 

PRINCIPLES    OF    GUSTAV 

RITTER  VON  WEX 


BY 

DAVID   A.  MOLITOR,  C.E. 

MEM.  AM.  Soc.  OF  C.E.;  MEM.   NATIONAL  GEOGRAPHIC  Soc.; 

MEM.  Ass.  FOR  ADV.  OF  SCIENCE;  MEM.  Soc.  PROMOTION  ENG.  EDUCATION; 

DESIGNING  ENGINEER,  ISTHMIAN  CANAL  COMMISSION 


FfRST  EDITION 

FIRST    THOUSAND 


NEW   YORK 

JOHN   WILEY   &   SONS 

LONDON:   CHAPMAN   &  HALL,  LIMITED 

1908 


COPYRIGHT,  1908, 

BY 

DAVID  A.  MOLITOR 


5tanbope  ipress 

F.    H.   GILSON     COMPANY 
BOSTON.     U.S.A. 


To  THE  MEMORY  OF 
GUSTAV   RITTER  VON  WEX 

THIS    WORK    IS 
DEDICATED 

Hofrat  Gustav  Ritter  von  Wex 

was  aulic  counsellor  and  chief  director  of  the 

Danube  River  regulation  and  improvement  at  Vienna 

knight  of  several  imperial  orders  ;  and  member 

of  many  scientific  societies.     He  was  born 

1811  and  died  Sept.  26,  1892, 

in  Ischl,  Austria. 


iii 


1 96462 


PREFACE 

IT  seems  strange  that  the  earnest  efforts  of  so  high  a  technical 
authority  as  Hofrat  von  Wex  should  have  failed  to  interest  hydrau- 
lic engineers  the  world  over.  A  careful  search  through  the  lead- 
ing hydraulic  literature,  with  one  exception,  did  not  reveal  a  single 
comment  regarding  either  the  man  or  his  work.  Prof.  I.  P. 
Church  says  in  his  "Mechanics  of  Engineering,"  1906  ed.,  foot 
of  page  688,  "  Herr  Ritter  von  Wex  in  his  Hydrodynamik  derives 
formulae  for  weirs,  in  the  establishing  of  which  some  rather  peculiar 
views  in  the  mechanics  of  fluids  are  advanced." 

The  unquestioned  ability  of  Hofrat  von  Wex  and  his  very  exten- 
sive practical  experience  along  the  lines  he  has  treated,  place  his 
work  in  the  front  rank  of  technical  achievement  in  the  specialty  of 
river  hydraulics.  His  views  and  theories  on  this  subject,  while 
radically  different  from  those  of  his  time,  are  most  rational  and 
sound,  and  merit  the  respect  and  approval  of  all  practical  hydraulic 
engineers. 

With  a  thorough  conviction  of  the  high  value  of  the  Wex  theories, 
the  author  has  ventured  to  place  them  before  his  profession  in  a 
form  which  he  hopes  will  prove  most  practical  and  acceptable. 

The  general  status  of  our  knowledge  respecting  hydraulics 
generally,  and  particularly  the  subject  of  weirs,  is  very  unsatis- 
factory, to  say  the  least.  Hence  any  progress,  if  it  be  real  progress, 
means  a  radical  departure  from  former  conceptions. 

It  is  the  rule  that  new  and  progressive  ideas  are  received  with 
more  or  less  suspicion,  which  Professor  Church  expresses  in  the 
word  "peculiar."  As  such  ideas  become  more  generally  known 
they  receive  a  more  charitable  reception  and  are  soon  tolerated. 
When  all  opposition  fails  they  are  crowned  by  final  acceptance. 
May  the  present  work  advance  to  this  final  state. 


vi  PREFACE 

At  this  age  of  water  power  development,  this  little  book  should 
enjoy  a  hearty  welcome.  It  was  prepared  with  the  utmost  care, 
and  while  theoretical  in  its  nature,  it  was  written  for  the  practical 
man.  Simplicity  and  clearness  were  the  first  requirements,  fol- 
lowed by  a  logical  and  practical  arrangement  in  the  presentation 
of  the  subject. 

It  was  not  deemed  advisable  to  enlarge  the  work  by  the  addi- 
tion of  mathematical  tables  as  is  usually  done.  On  the  contrary, 
such  tables  are  of  little  value  when  dealing  with  general  problems 
and  in  the  author's  opinion  the  most  useful  aid  to  the  solution  of 
the  formulae  here  given  is  a  copy  of  Barlow's  tables  of  squares, 
cubes,  square  roots  and  cube  roots,  and  Zimmermann's  Rechen- 
tafel,  being  a  multiplication  table  of  all  numbers  from  i  to  1000  by 
all  numbers  from  i  to  100.  These  universal  tools  should  occupy 
a  prominent  place  on  every  engineer's  book  shelf,  and  nothing 
better  can  be  proposed  here  as  labor  saving  devices. 

Attention  is  called  to  the  valuable  information  collected  in 
Appendix  A,  which  constitutes  a  most  complete  exposition  of  all 
known  older  formulae  for  overfalls.  This  in  itself  is  the  best 
argument  which  can  be  presented  in  defense  of  the  new  formulae 
to  which  this  work  is  devoted. 

Appendix  B  contains  the  solution  of  a  novel  problem  in  Hy- 
draulics, which,  so  far  as  known,  has  never  heretofore  been  solved 
in  any  satisfactory  manner.  The  solution  there  given  is  theoretically 
correct,  and  probably  more  accurate  than  the  knowable  accuracy 
of  the  empiric  coefficients  would  really  justify. 

In  Appendix  C,  all  the  new  formulae  are  arranged  in  tabulations 
for  ready  reference,  thus  avoiding  loss  of  time  in  picking  out  special 
cases  from  the  text.  It  is  believed  that  this  offers  a  very  attractive 
summary  of  the  most  useful  contents  of  the  book. 

The  author  here  wishes  to  acknowledge  his  indebtedness  to 
Herr  Wilhelm  Engelmann,  of  Leipzig,  Germany,  for  many  favors 
extended  and  advice  given  prior  to  undertaking  the  present  work. 
Also  to  Professor  Gardner  S.  Williams  and  Mr.  Allen  Hazen, 
members  Am.  Soc.  C.  E.,  for  their  kind  permission  to  use  the 


PREFACE  vii 

tabulations  relating  to  effect  of  weir  crests  as  published  in  their 
"Hydraulic  Tables,"  pp.  71-75. 

In  conclusion  he  wishes  to  express  his  obligation  and  thanks  to 
Mr.  Alex.  Ilich  Wolkowyski,  C.  E.,  Ass't  Eng'r,  Isthmian  Canal 
Commission,  for  valuable  assistance  rendered  in  the  preparation 
of  this  work,  also  to  Messrs.  John  Wiley  and  Sons  for  the  most 
excellent  manner  in  which  they  have  accomplished  the  publication. 

DAVID    A.    MOLITOR. 

WASHINGTON,  D.C.,  December  5,  1907, 


CONTENTS 

PAGE 

INTRODUCTION  .  i 


CHAPTER   I. 


FUNDAMENTAL  EQUATIONS  .    .    . 
Flow  through  Lateral  Orifices. 


CHAPTER   II. 

COMPLETE  OVERFALL  WEIRS u 

A.  Derivation  of  new  formulae  for  the  discharge  over  complete 
overfall  weirs,  built  normally  to  the  direction  of  the  current. 

B.  Derivation  of  new  formulae  for  the  discharge  over  complete 
•    overfall  weirs,  built  obliquely  to  a  channel  or  represented  in 

plan  by  a  curved  or  broken  line. 


CHAPTER   III. 

INCOMPLETE  OVERFALL  WEIRS 28 

Derivation  of  new  formulae  for  discharge  over  incomplete 
overfalls  or  submerged  weirs,  and  through  contracted  river 
channels. 

CHAPTER   IV. 

SLUICE  WEIRS  AND  SLUICE  GATES 36 

Derivation  of  new  formulae  for  discharge  over  sluice  weirs 
and  through  sluice  gates  for  regulating  works.  Three  cases. 

CHAPTER  V. 

BACKWATER  CONDITIONS 46 

Discussion  and  formulae  for  backwater  height  and  distance; 
also  computations  of  the  dimensions  of  weirs  for  obstructed 
flow  when  the  quantity  of  flow  and  backwater  height  are 
fixed.  Illustrated  by  problems. 


x  CONTENTS 

CHAPTER  VI. 

PAGE 

FLOW  IN  RIVERS  AND  CANALS 52 

Derivation  of  formulae  for  discharge  from  rivers  or  lakes 
into  water  power  canals  or  flumes,  i.  General  Discussion. 
2.  Proposed  general  solution.  3.  When  all  the  available 
water  is  to  be  diverted.  4.  When  only  a  portion  of  the  avail- 
able water  is  to  be  diverted,  and  the  remainder  to  be  discharged 
over  a  weir  built  normally  to  the  river.  5.  The  same  as  the 

previous  case  for  a  weir  built  diagonally  to  the  river. 

* 

CHAPTER   VII. 

EMPIRIC  COEFFICIENTS 75 

i.  Introductory.  2.  Complete  overfalls ;  a,  weirs  normal  to  the 
channel  and  no  wing  walls,  and  b,  weirs  normal  to  the  channel 
but  contracted  on  the  ends  by  wing  walls.  3.  Incomplete 
overfalls.  4.  Sluice  weirs  and  gates.  5.  End  contractions. 
6.  Weir  crests. 

APPENDIX  A. 

A  COLLECTION  OF  WEIR  FORMULAE  PROPOSED  BY  DIFFERENT 
AUTHORS. 

1.  COMPLETE  OVERFALLS 102 

2.  INCOMPLETE  OVERFALLS „ 109 

APPENDIX  B. 

ON  THE  FLOW  OVER  A    FLIGHT    OF    PANAMA    CANAL   LOCKS. 
A  NOVEL  HYDRAULIC  PROBLEM. 

APPENDIX   C. 

A  TABULATION  OF  THE  NEW  FORMULAE, 
arranged  for  ready  reference. 


DEFINITIONS  OF  TERMS  USED  THROUGH- 
OUT THIS  WORK 

A  =  discharge  area  in  square  feet. 

Q  =  discharge  quantity  in  cubic  feet  per  second. 

v  =  mean  velocity  of  approach  in  feet  per  second. 

V  =  mean  velocity  of  discharge  in  feet  per  second. 

S  =  the  unit  pressure  at  any  point  of  the  discharge  area. 

iSj  and  S2  are  special  values  of  S. 

g  =  acceleration  due  to  gravity,  in  feet  per  second  =  32.09  at 
the  equator,  and  32.26  at  the  pole,  both  at  sea  level. 

7  =  the  weight  of  a  cubic  foot  of  water,  usually  taken  as  62.5 
pounds. 

n  =  Dubuat's  coefficient  =  0.67. 

//  =  empiric  coefficient  for  discharge  into  free  air. 
/£t  =  empiric  coefficient  for  submerged  discharge. 

T  =  total  depth  of  approach  channel  in  feet. 
Tl  =  total  depth  of  discharge  channel  in  feet. 
H  =  depth  of  flow  of  approach  over  crest  of  weir. 
Hl  =  depth  of  flow  of  discharge  over  crest  of  weir. 
H2  =  H  —  it^  =  difference    between    the    approach    and    dis- 
charge surfaces. 

k  =  depth  of  water  on  upstream  side  of  weir  and  below  weir 
crest. 

B  =  width  of  approach  channel. 

b  =  width  of  the  discharge  section,  or  length  of  weir  crest. 

b'  =  width  of  a  diversion  channel. 

d  =  depth  at  entrance  to  a  diversion  channel. 


xii  TERMS  USED  THROUGHOUT  THIS  WORK 

-^  =  angle  of  inclination  of  the  upstream  face  of  a  weir  dam  with 
the  horizontal. 

$  =  angle  of  inclination  of  any  wing  dam  with  the  side  of  the 

channel. 
0  =  angle  of  inclination  of  a  diversion  channel  with  the  main 

channel. 

s  =  surface  slope  =  fall  divided  by  length,  both  in  feet. 

^ 
r    =  mean  hydraulic  radius  =  —  . 

w 

w  =  wetted  perimeter. 

D  =  original  depth  of  any  river,  previous  to  placing  any  obstruc- 
tion in  its  course. 

Z  =  total  backwater  height,  measured  above  the  original  or 
natural  surface  slope. 

L  =  total  backwater  distance,  measured  from  the  crest  of  the 
weir  or  dam  obstruction. 

/  and  z  =  co-ordinates  of  backwater  surface  referred  to  the  original 
slope  as  the  /-axis  and  origin  in  the  vertical  through  the 
crest  of  the  weir. 

C  =  either  Bazin's  or  Kutter's  coefficient  for  the  Chezy  formula. 


OF   THE 

UNIVERSITY 

OF 


HYDRAULICS 
OF   RIVERS,   WEIRS   AND    SLUICES 


INTRODUCTION 

IN  treating  a  subject  of  this  nature  it  is  very  important  that  the 
language  used  should  be  clear  and  specific.  To  accomplish  this 
end  a  few  of  the  most  important  terms  must  be  accurately  defined 
so  that  there  may  be  no  doubt  as  to  the  particular  meaning  implied 
by  them.  Much  confusion  is  often  created  by  a  promiscuous  use 
of  technical  terms,  especially  when  these  have  received  a  variety 
of  definitions  by  different  authors.  The  following  definitions  will 
be  adhered  to  throughout  the  present  work. 

Hydrostatic  pressure  is  the  pressure  exerted  by  the  weight  of  a 
column  of  water  and  acts  with  equal  intensity  in  all  directions. 
It  implies  pressure  due  to  water  in  a  state  of  rest. 

Hydrodynamic  pressure  is  that  produced  by  a  stream  or  jet  of 
water  impinging  on  a  surface,  and  may  be  less  than,  equal  to  or 
greater  than  the  hydrostatic  pressure.  It  is  a  function  of  velocity 
only.  It  is  always  expressed  as  a  static  head  v*/2g. 

Hydraulic  pressure  is  the  resultant  water  pressure  on  any  sur- 
face caused  by  any  possible  combination  of  both  hydrostatic  and 
hydrodynamic  pressures. 

Velocity  in  all  of  the  relations  here  considered  will  always  be 
regarded  as  the  mean  velocity,  being  equal  for  all  points  of  the  same 
section  normal  to  the  direction  of  flow.  Hence  the  quantity  of 
discharge  passing  a  given  section  A  per  second  of  time  will  be 
Q  =  Av. 

Velocity  of  approach,  v,  is  the  mean  velocity  at  such  a  section  of 
an  approach  channel  where  the  effect  of  an  overfall  is  still  too 
small  to  produce  an  acceleration. 


2  INTRODUCTION 

Velocity  of  discharge,  V,  is  the  mean  velocity  at  such  a  section  of 
a  discharge  channel  where  the  flow  is  again  uniform  or  normal 
after  having  been  accelerated  by  passing  over  an  overfall. 

An  overfall  is  a  vertical  drop  in  the  bottom  of  a  channel  and 
may  be  complete  or  incomplete  accordingly  as  it  is  or  is  not  sub- 
merged below  the  lower  water  level  or  lower  pool. 

A  complete  overfall  weir  is  a  weir  the  crest  of  which  is  above 
the  level  of  the  lower  pool. 

An  incomplete  overfall  weir,  usually  called  a  submerged  weir,  is 
such  a  weir  the  crest  of  which  is  below  the  level  of  the  lower 
pool. 

A  sluice  weir  is  a  weir  the  flow  over  which  is  partially  obstructed 
by  a  gate,  thus  producing  a  condition  of  flow  resembling  that 
through  a  lateral  orifice  in  the  side  of  a  vessel. 

The  ends  of  a  weir  are  usually  the  side  walls  of  the  channel. 
The  end  walls  will  be  supposed  to  be  vertical  while  the  front  and 
back  of  the  weir  or  dam  may  have  various  dimensions.  The  crest 
of  the  dam  may  have  a  variety  of  forms  which  are  given  special 
consideration  in  Chapter  VII. 

Contraction  is  a  term  used  to  designate  the  diminution  in  the  flow 
area  just  beyond  the  discharge  area  when  the  discharge  proceeds 
into  free  air  and  is  not  confined  after  leaving  the  discharge  section. 

Complete  contraction  is  contraction  around  the  entire  periphery 
of  the  discharge  section. 

Partial  contraction  is  contraction  on  only  one,  two  or  three  sides 
of  the  discharge  section. 

End  contraction  is  contraction  on  the  two  ends  of  a  weir.  This 
occurs  only  when  the  side  walls  of  the  channel  are  suddenly  ended 
at  the  vertical  through  the  weir  crest. 

All  the  following  formulae  are  derived  without  reference  to  end 
contractions  or  shape  of  weir  crest,  and  the  manner  of  dealing  with 
these  special  features  is  discussed  in  Chapter  VII. 

In  the  several  chapters,  I  to  VI,  the  new  formulae  for  a  great 
variety  of  weirs  and  sluices  were  derived  and  these  were  finally 
tabulated  in  usable  form  in  Appendix  C.  Chapter  VII  is  devoted 


INTRODUCTION  3 

to  a  determination  of  empiric  coefficients  for  the  new  formulae 
and  these  results  are  likewise  included  in  the  tabulations  of  Appen- 
dix C. 

The  inconsistencies  and  irrational  constitution  of  the  older  for- 
mulae are  discussed  at  some  length  in  Appendix  A,  to  which  special 
attention  is  called  here  as  furnishing  the  real  justification  for  the 
new  formulae.  It  was  not  deemed  desirable  to  interrupt  the 
continuity  of  the  argument  by  discussing  old  formulae  at  every 
opportunity,  and  hence  these  were  collected  into  an  Appendix. 
In  this  way  the  reader  will  better  understand  the  criticisms  after 
having  become  familiar  with  the  subject  matter  of  the  book. 


CHAPTER    I. 

FUNDAMENTAL   EQUATIONS. 
Flow  through  Lateral  Orifices. 

THE  fact  that  the  fundamental  equations  for  flow  through  lateral 
orifices  have  been  applied  by  many  hydraulicians,  without  proper 
modification,  to  determine  the  flow  through  canals  and  rivers, 
contracted  by  the  introduction  of  weirs  or  sluices,  makes  it  desir- 
able to  review  the  derivation  of  these  equations,  and  to  show  that 
such  application  is  not  justifiable. 

The  fundamental  equations  for  the  flow  of  water  through  lateral 
orifices  in  the  vertical  sides  of  a  large  reservoir,  in  which  the  water 
is  perfectly  quiet,  and  is  maintained  at  a  constant  level,  will  be 


derived  in  the  following.     These  equations  are  generally  accepted 
and  will  be  used  in  the  derivation  of  the  new  weir  formulae. 

In  Fig.  i,  A  BCD  represents  the  vertical  wall  of  the  reservoir 
with  the  orifice  EFGH,  through  which  the  water  flows  freely  into 
the  air.  According  to  the  principle  of  hydrostatics,  the  pressures 
at  any  _ppints,  J  or  G  in  this  orifice,  are  equal  to  the  columns  of 
water  EJ  and  EG,  respectively,  and  it  has  been  experimentally 


FUNDAMENTAL  EQUATIONS  5 

established  that  the  velocity  with  which  the  water  flows  through 
small  openings  at  /  or  G,  is_equal  to  the  velocity  of  a  body  falling 
in  air  through  the  heights  EJ  and  EG,  respectively.  Hence,  the 
velocity  at  the  point  /  is  JK  =  V  2  gEJ3  and  the  velocity  at  the 
point  G,  is  GL  =V2  gEG,  in  which  g  =  acceleration  of  gravity 
per  second  of  time. 

Using  notation  indicated  in  Fig.  i,  and  calling  x  the  ordinate  of 
a  filament  of  water  ad)  of  height  dx,  and  flowing  with  a  velocity  y, 
then  the  differential  equation  representing  the  quantity  of  water 
in  this  filament  is 

dQ  =  b  .dx  .  V2  gx. 

To  obtain  the  total  quantity  of  water  flowing  through  the  orifice 
per  second,  this  equation  must  be  integrated  between  the  limits 
x  =  o  and  x  =  H,  giving 

f*B  _  _     f*H 

Q  =    I      bdx  \/2  gx  =b  \/2  g  I     x*  dx, 
J  J 


or  Q  =  f  x*b  V~g  +  C. 

For  x  =  o,  Q  becomes  zero,  hence  the  constant  C  is  zero. 
Therefore,  for  x  =  H,  the  quantity  of  flow  through  the  orifice 
b  .  H,  per  second,  becomes 

Q  -  §  W?1  V~g  -  }  bHV^H, 
in  which  b  .  H  equals  the  area  of  orifice,  and 
f  \/2  gH  =  mean  velocity. 

Since  the  values  of  y  vary  as  the  square  root  of  the  corresponding 
values  of  x,  the  curve  EKL  is  a  parabola. 

However,  experiments  have  shown  that  the  friction  existing 
between  the  particles  of  water  among  themselves  and  at  the  edges 
of  the  orifice,  causes  a  retardation  in  this  theoretical  velocity. 
Hence,  the  actual  quantity  of  flow  will  always  be  less  than  the 
above,  and  the  true  equation  must  contain  an  empiric  coefficient 
to  correct  for  the  combined  effect  of  friction  and  contraction.  The 


6  HYDRAULICS 

equation  of  flow  through  a  rectangular  orifice  in  the  vertical  side 
of  a  reservoir  in  which  the  water  is  perfectly  quiet  and  retained  at 
a  constant  level  may  then  be  written  thus: 


Q  =  $f*bHV2gH  =  %f*bV2gH*    .     .     .     (i) 

Should  the  orifice  EFHG  be  partly  closed  by  a  sluice  gate, 
EFRJ,  then  the  quantity  of  flow  through  the  remaining  orifice, 
JRHGj  will  be  the  total  quantity  as  from  Eq.  (i),  less  the  quantity 
which  is  prevented  from  flowing  out  by  the  gate.  This  quantity 
is  represented  by  the  body  JRHGKPNL,  Fig.  i,  and  calling  the 
height  EJ  =  Hlt  equation  (i)  when  applied  to  this  case  becomes 

Q=it*bV^~g(H*  -H*)      ....     (2) 

Should  the  sluice  gate  be  lowered  to  a  line  ik,  leaving  only  an 
orifice  of  breadth  b  and  height  a,  which  latter  is  small  in  comparison 
to  the  height  H,  the  water  may  be  assumed  to  flow  through  such 

an  orifice  under  an  average  pressure  ( H  — J  and  the  quantity  of 
flow  per  second  will  become 


Q=l^ab\j2g\H-^ (3) 

These  three  fundamental  equations  have  been  derived  by  most 
hydraulicians,  and  their  correctness  having  been  established  by 
many  accurate  experiments,  they  may  be  generally  accepted. 
However,  it  must  not  be  forgotten  that  these  equations  apply  only 
to  cases  in  which  the  orifice  is  very  small  compared  to  the  size  of 
the  reservoir,  so  that  the  water  in  the  latter  may  retain  its  height 
and  remain  unagitated. 

Before  passing  on  to  the  derivation  of  the  weir  formulae,  it  will 
be  advisable  to  make  an  exposition  of  facts  which  have  been  gen- 
erally neglected  by  other  authors  in  treating  of  submerged  weirs. 

Let  Fig.  2  represent  the  longitudinal  section  of  a  river  in  which 
isjplaced  a  submerged  weir  LM  having  its  crest  below  the  surface 
FG  by  an  amount  Hl  and  damming  the  water  to  a  height  H 


FUNDAMENTAL  EQUATIONS  7 

above  the  crest  of  the  weir.  The  quantity  of  water  Q,  passing  over 
this  weir  per  second,  is  regarded  as  being  made  up  of  two  parts, 
that  flowing  into  the  air  through  the  upper  portion  EK  of  the  entire 
section  EL,  which  may  be  found  from  Eq.  (i);  and  that  flowing 


Fig.  2 


through  the  submerged  area  KL  under  the  uniform  pressure  head 
H  —  Hr  This  is  the  basis  of  the  argument  generally  applied  in 
deriving  weir  formulae  and  leads  to  very  erroneous  results,  as  will 
presently  be  shown. 

Regarding  the  first  of  these  increments  of  quantity,  the  following 
criticism  is  offered.  As  the  channel  leading  up  to  a  weir  is  always 
of  limited  dimensions,  and  the  weir  and  other  possible  obstructions, 
such  as  wing  dams,  etc.,  must  affect  the  conditions  of  flow  through 
the  upper  part  EK  of  the  section  EL,  Fig.  2;  and  since  the  theo- 
retical Eq.  (i),  when  applied  to  this  flow,  does  not  involve  in  any 
way  the  dimensions  of  the  reservoir,  or  the  channel  dimensions  in 
the  case  of  weirs,  it  follows  that  the  theoretical  equation  cannot 
be  adapted  to  the  weir  condition  by  the  mere  introduction  of  an 
empiric  coefficient.  It  is  also  assumed  that  the  velocity  of  approach 
exerts  an  hydraulic  pressure  only  on  the  area  of  flow,  while  it  is 
positively  known  that  this  velocity  likewise  affects  the  surface  of 
the  weir,  and  all  other  surfaces  of  the  channel  approaching  the 
weir.  These  pressures  are  deflected  into  the  area  of  flow  in  a 
manner  dependent  on  the  shape  of  weir  and  other  parts  of  the 
channel. 

In  regard  to  the  second  increment  of  flow,  it  is  assumed  that 
the  lower  pool  exerts  a  back  pressure  on  the  part  section  KL,  the 


8 


HYDRAULICS 


same  as  if  the  water  was  not  in  motion.  To  prove  that  this  is 
not  in  accordance  with  the  existing  conditions,  the  following 
experiments  are  cited. 

Referring  to  Fig.  3,  in  which  water  flows  freely  through  the 
irregular  vessel  ABCD,  and   the  flow  is  supplied    from  a  large 
reservoir,  so  that  the  level  MMt  remains  con- 
jWt    stant,  then  the  resultant  hydraulic  pressure  at 
any  point  of  the  vessel  is  equal  to  the  hydro- 
static pressure  at  that  point,  less  the  velocity 
height  of    the  water   flowing  past  this  same 
point.     (See  Ruehlmann,  Hydromechanik,  pp. 
211  and  214.) 

If  in  a  contracted  section  EF,  the  water 
flows  with  a  very  high  velocity  vl}  such  that 

FiS-  3  the  velocity  height  —  becomes    greater    than 

the  hydrostatic  head  h±  at  a  certain  point  F,  then  the  resulting 
hydraulic  pressure  on  the  above  hypothesis  becomes  negative  and 

equal  to  (hl  —  — V  indicating  a  suction.     Now,  if  a  glass  tubeFJ 

be  connected  with  the  vessel  at  JP,  then  the  water  contained  in  a 
vessel  KL  will  be  drawn  up  into  the  tube  by  an  amount 

ab  =  ( -1 h1 } .    Also,  since  the  pressure  at  any  section  EF  must 

\2  g         I 

be  equal  in  all  directions,  it  is  apparent  that  the  same  suction  would 
be  produced  if  the  opening  of  the  tube  were  on  the  lower  side  at  e. 
Hence,  the  statement  may  be  made  that  for  the  conditions  of  flow 
just  described,  the  water  flowing  past  an  orifice  e  in  the  lower  side 
of  a  tube  FJ,  will  produce  a  suction  in  this  tube  equal  to 


g 

To  determine  the  force  of  impactof  water  flowing  with  velocity  v 
through  a  flume  against  a  disc  CD,  Fig.  4  (which  question  has 
not  yet  been  satisfactorily  solved),  Dubuat  made  numerous  experi- 
ments and  found  that  the  hydraulic  pressure  on  the  back  face  of 


FUNDAMENTAL  EQUATIONS  9 

the  disc  CD  is  equal  to  the  hydrostatic  head  h  on  the  surface,  less 

v2 
0.67 —  ,  proving  that  in  this  case  there  is  also  a  suction  on  the 


Fig.  4 

V2 

surface  CD  equal  to  the  velocity  height  o.  67 (See  Ruehlmann, 

2  g 
Hydromechanik,  p.  596.) 

The  suction  in  this  latter  case  is,  however,  less  than  for  the 
closed  vessel,  which  is  supposed  to  be  due  to  the  difference  between 
a  closed  vessel  and  an  open  channel,  and  also  that  the  eddies  pro- 
duced behind  the  disc  exert  a  certain  impact  opposing  the  suction 
and  thereby  diminishing  the  latter. 

Darcy  found,*  by  experimenting  with  a  Pitot  tube,  that  when 
the  orifice  was  pointed  perpendicular  to  the  direction  of  flow,  the 
hydrostatic  column  in  the  tube  was  lowered  by  an  amount 

v2 
h2  =  0.678 — -below  the  surface  of  the  water,  this  being  the  result 

o 

of  suction  produced  by  the  water  in  flowing  past  the  orifice.  When 
the  orifice  was  pointed  with  the  current  this  suction  amounted  to 

v2 
only  h3  =  0.434 —  .     Since  the  suction  in  the  latter  case  should 

undoubtedly  be  greater  than  in  the  former,  it  seems  reasonable  to 
suppose  that  in  the  second  case  the  filaments  of  water  are  deflected 
by  the  tube  in  such  manner  as  to  diminish  the  suction  effect  on  the 
orifice  when  the  tube  is  pointed  with  the  current. 

The  fact  that  the  value  of  the  amount  of  suction  found  by  Darcy 
on  the  Pitot  tube  is  less  than  was  obtained  for  the  flow  through  the 
vessel  in  Fig.  3,  and  less  than  was  found  by  Dubuat  for  the  disc, 

*  See  Ruehlmann,  Hydromechanik,  p.  383. 


10  HYDRAULICS 

Fig.  4,  is  probably  due  to  the  friction,  cohesion  and  capillarity  to 
be  expected  by  the  flow  along  the  small  conical  pressure  tube  used. 
Until  better  experiments  on  these  lines  shall  have  become  avail- 
able, the  results  of  Dubuat  as  o.  67  —  may  be  safely  accepted. 

Since  the  above  experiments  make  it  apparent  that  there  is  a 
suction  on  the  lower  surface  of  all  incomplete  weirs,  submerged 
weirs,  and  sluice  gates,  which  suction  diminishes,  the  hydrostatic 
counterpressure  of  the  part  section  KL,  Fig.  2,  by  an  amount 

i? 
0.67 — •  ,  it  follows  that  a  larger  quantity  of  flow  is  permitted 

through  the  submerged  section,  than  is  assumed  under  the  supposi- 
tion that  the  water  in  the  lower  pool  is  perfectly  quiet  and  exerts 
an  hydrostatic  pressure  on  the  submerged  section  over  its  entire 
height  KL. 

Hence,  the  generally  accepted  basis  for  weir  formulae  is  erroneous, 
and  in  the  following  chapters  new  formulas  are  derived  on  the 
basis  of  more  rational  assumptions.  Complete  overfall  weirs  are 
treated  first  as  a  matter  of  convenience. 


CHAPTER   II. 
COMPLETE  OVERFALL  WEIRS. 

A.    Derivation  of  New  Formula  for  the  Discharge  over  Com- 
plete Overfall  Weirs  built  normally  to  the  Direction  of  the  Current. 

THESE  formulae  are  derived  for  the  following  conditions,  viz.: 
That  the  water  reaches  the  weir  section  with  a  certain  initial 


Fig.  5 

velocity  v\  that  all  the  water  in  the  channel  must  flow  over  the  weir; 
that  the  weir  is  horizontal  and  has  an  inclined  upstream  face; 


5 

k 

--> 

\ 

i'  , 

F 

' 

^' 

, 

Velocity  v  per  sec.                     Ps 

Quantity  Q  per  sec. 

-''''   \h    ^> 

e*\  ^&  m* 

Fig.  6 

and  that  the  direction  of  the  weir  be  normal  to  the  direction  of 
flow.  In  the  general  case,  a  wing  dam  is  assumed  located  on  each 
side  of  the  weir.  (See  Figs.  5  and  6.) 

ii 


12  HYDRAULICS 

Let    g=  acceleration  of  gravity. 
ft=  coefficient  of  flow. 
7=  weight  of  i  cubic  foot  of  water. 
Other  notations  as  per  diagrams. 

The  breadth  of  the  weir  is  &;  that  of  the  channel  is  E\  the  wing 
walls  extend  above  the  water  level. 

From  the  dimensions  and  depth  of  water 

Q  =  B  (H  +  k)  v  =  BTv    ........     (4) 

v=          = 


The  projected  length  of  each  wing  wall  on  the  direction  of  the 

B-b 

weir  is  -  . 
2 

The  water  flowing  over  the  weir  takes  the  surface  curve  COM, 
but  the  quantity  of  flow  is  the  same  as  if  the  surface  were  GNM, 
the  effective  fall  being  the  same  in  each  case. 

The  forces  acting  on  the  entire  structure  and  those  acting  on 
the  discharge  area  will  now  be  determined. 

1.  The  hydrostatic  pressure  on  the  section  bH  is 

rir>£ce 

*,-7>f  -      ......    (6) 

2.  The  hydrodynamic  pressure,  equally  distributed  over  the  area 
bH,  and  resulting  from  a  prism  of  moving  water  of  area  bH  and 
velocity  v,  is 


3.    The  hydrodynamic  pressure  against  a  -fixed  surface  is  found 

v2  v 

to  be  =  yF  -  =  yq  -  ........    (8) 

g  g 


COMPLETE  OVERFALL  WEIRS  13 

by  Weisbach's  experimental  law*  viz.:  "When  water  flows  in  a 
channel  enclosed  by  three  sides,  the  impact  against  a  fixed  surface 
in  the  channel  will  be  equal  to  the  impact  of  an  isolated  stream  of 
water  of  same  cross-section  as  the  water  in  the  channel.  As 


found  from  the  figure,  the  value  of  q  in  Eq.  (8)  is  v  I—     —\H, 

and  the  cross-section  of  the  water  in  the  channel  is  ( }H  =  F. 

V      2      I 


This  would  indicate  that  the  hydraulic  pressure  against  a  fixed 
surface  would  be  twice  that  of  a  stream  flowing  against  an  opening 
of  same  cross-section. 

But  since  in  the  above  case  the  water  is  not  confined,  and  is  thus 
easily  deflected  towards  the  area  of  flow,  the  quantity  q  must  be 
assumed  for  the  latter  case.  Hence  the  pressure  against  each  wing 


v 
wall  =  P    =  y    -- 


2  g  2g 


IB  -  b\ 
-      -       .......    (9) 

\       2       / 


which  force  may  be  assumed  to  act  in  the  center  of  gravity  of  the 
obstructed  prism. 

Regarding  the  divergence  of  this  water,  it  should  be  considered 
that  the  filaments  passing  along  the  sides  of  the  canal  must  be 
deflected  through  the  angle  <£,  while  those  passing  along  the  line 
LF  are  not  deflected  appreciably;  hence,  the  angle  of  divergence 

for  the  entire  prism  is  taken  as—. 

2 

The  force  Ps_==_eJ  (see  Fig.  6)  may  be  resolved  into  the  com- 
ponents //  and  nf,  the  former  representing  the  pressure  which  is 
effective  against  the  discharge  area,  and  the  latter  that  expended 
on  the  sides  of  the  canal  and  wing  walls. 

From  Fig.  6, 

Tj  =  ejcos  -  =  P3  cos  -  =yq—  cos  -. 

2  2  2  g  2 

*  Weisbach  Experimental  Hydraulik,  §  423.  —  Ed. 


14  HYDRAULICS 

The  quantity  thus  reaching  the  weir  cannot  continue  its  flow 
in  this  direction  and  is  again  deflected  into  the  direction  W,  hence, 


(10) 


which  represents  the  pressure  effectively  expended  against  the 
discharge  area,  while  the  force  if  produces  a  contraction  of  the 
flow  in  the  flow  area. 

Therefore,  the  total  pressure  against  the  discharge  area  from  the 
quantity  of  water  partially  obstructed  by  the  two  wing  walls  is 

cos2     .          (ii) 


g  2  g  \       2  2 

This  pressure  may  be  represented  by  a  volume  of  area  bH  and 
length  t,  thus  P4  =  ybHd  from  which 

P.         v*  IB  -  b\       2  <£ 

t==~T^r  =  T\~     -    cos2^.      .      .      .     (12) 
ybH      bg\     2    /          2 

4.  The  pressure  resulting  from  the  volume  of  -water  q^  =  Bkv  in 
the  lower  channel  area  Bk  is  now  found. 

Since  this  water  q^  flows  in  a  three-sided  channel  its  hydro- 
dynamic  pressure  on  the  sloped  weir  area  may  be  taken  as 

p*-ni--1-.Bk     .....    (13) 

o  o 

which  pressure  may  be  regarded  as  acting  in  the  axis  of  the  channel. 
The  reasoning  previously  applied  gives  for  this  case  an  average 

angle  of  divergence  for  this  water  equal  to  -  . 

Resolving  P5  into  components  ro2  and  uo2,  Fig.  5,  the  latter  being 
effective  on  the  discharge  area,  it  is  found  from  the  figure  that 

no*  =  0,0,  cos  -  =  P5  cos-  =  7  -  Bk  cos  —    .     .    (14) 
2  2  2 


COMPLETE  OVERFALL  WEIRS  15 

Since  this  force  o2w  tends  to  lift  the  overlying  water  over  the 
weir,  it  must  be  deflected  horizontally  by  the  counteraction  of  the 

upper  strata,  thus  giving  the  resultant  PQ  =  WjW  =  u^w  cos  — 


=  P5  cos2  S  or  PQ  =  7  -  Bk  cos 
2  g 


(15) 


This  pressure  certainly  has  its  maximum  effect  on  the  discharge 
area  just  at  the  surface  of  the  weir,  which  effect  gradually  dimin- 
ishes until  it  becomes  zero  at  the  surface  of  the  water.  Hence, 
the  total  effective  result  of  PQ  on  the  area  bk  may  be  represented 
by  a  triangular  prism  of  length  b,  height  H  and  bottom  breadth  /?, 


thus: 


i/r 

— — -  cos2 * 
bgH          2 


(16) 


The  quantity  of  water,  of  depth  k,  striking  the  wing  walls,  can 
reach  the  flow  area  only  by  material  deflection  and  is  here  neglected 
as  being  small  and  introducing  too  many  complications. 


Fig.  7 


The  individual  pressures  Plt  Pv  P4,  and  Pe  are  now  combined 
(See  Fig.  7.) 

1.   The  area  OEE^  represents  P2  =  bH  —  by  making 

OE  =  JE,  =  —  • 


1  6  HYDRAULICS 

2.    Again,  by  making 


the  area  OO^R^  will  represent  the  pressure  P4  against  the  dis- 
charge area. 

3.  Also,  by  making  O^R.  =  s^  =  H,  the  area  of  the  triangle 

_  Ifjz 

O^s  will  represent  Px  =  jj.  -  • 

4.  Lastly,   by  making  sz  =  /?  =  —  —  —  cos2  —  the  area  O.sz 

bgH  2 

will  represent  P6  =  7/9  —  • 
2 

In  conclusion,  the  sum  of  the  flow  areas  just  designated  will 
make  the  area  EE^zOv  and  a  prism  of  this  area  and  the  length  b 
will  represent  the  total  resultant  pressure  on  the  discharge  area 
E^K 

Now  to  compute  the  velocity  from  the  pressure,  continue  the 
line  zOl  to  U,  and  call  y  the  hydrodynamic  pressure  at  W,  of  a 
filament  of  water  distant  X,  below  the  surface,  and  having  a 
velocity  F.  Also  call  the  surface  pressure  EOl  =  S,  and  the 
pressure  at  the  crest  of  the  weir  =  E^z  =  Sr 

Then  from  similarity  of  triangles  UEO^  and 


UE  :  EO,  =  UEl  :  E& 

SH 


or  Z:5=  (Z  +  H)  iS.orZ  = 


,  -  S' 


also  from  similarity  of  triangles  UWV  and  UE^z 


UW  :  VW  =  UEl  :  E& 
or  (Z  +  X)  :  y  =  (Z  +  H)  :  S.ory  =  S, 


Hence,  V  =  /*v  2  gy 


COMPLETE  OVERFALL  WEIRS  17 

The  quantity  dQ  flowing  through  an  element  doc  at  the  point  W 
with  velocity  V  and  breadth  b  will  be 


dQ  =  bVdx  =  pbdoc\J2gSi   — — -    .     .     .     (17) 


To  find  Q  it  is  necessary  to  integrate  Eq.  (17)  between  limits 
X  —  o  and  X  =  H,  and  obtain 


when  X  =  o,  then  Q  =  o,  hence  the  constant  becomes 


and  by  substituting  this  value  for  C  in  the  above  expression,  and 
also  substituting  H  for  X,  the  discharge  through  the  area  bH,  in 
cubic  feet  per  second,  is  obtained  as  follows : 


or          Q=vb\ji[(Z  +  H)-Zl (18) 

/     7T91    \ 

Now  substituting  for  Z  its  value  (  — J  and  reducing,  the  fol- 
lowing form  is  obtained : 

1i-Sl].    .    .    .     (19) 


Since  5  =  EO1}  and  5t  =  EjZ,  these  values  become,  in  accord- 
ance with  previous  deductions, 


18 


HYDRAULICS 


S  =* \-  —  f )  cos2  -  from  which,  by  substitution  for 

2g      bg  \    2    /         2 

Q 


and 


H 


/ 2_Y 

\B  (k  +  H)/ 


cos^ 


-  .  (20) 


I.    For  a  straight  weir  perpendicular  to  the  channel  and  with 
vertical  face,  Eqs.  (19)  and  (20)  become 


bgH  \B  (k  +  H)/  ' 


.    .     .     (21) 


II.    For  same  weir  without  wing  walls,  whence  B  =  b, 
S~-'-(k  +  H))' 


gH 


•       (22) 


III. 


B  =  b-,k  =  o;(f>  =  o°;  and  ^  =  o°. 


•       (23) 


COMPLETE  OVERFALL  WEIRS  19 

which  is  Weisbach's  formula  for  complete  overfall  weirs  in  rivers. 
This  coincidence  is  important  as  it  proves  conclusively  that  Weis- 
bach^s  formula  for  complete  overfall  weirs  is  applicable  only  to  the 
case  shown  in  Fig.  8,  in  which  the  weir  offers  absolutely  no  obstruc- 
tion to  the  flow.  Such  weirs  are  not  built. 


Fig.  8 


IV.  When  a  portion  of  the  flow  is  diverted  through  a  lateral 
channel.  In  the  preceding  formulae  it  is  assumed  that  all  of  the 
approaching  water  must  flow  over  the  weir.  Should  a  portion  Qf 
of  this  water  be  wasted  through  a  lateral  outlet  or  over  a  weir, 
this  quantity  must  first  be  found  by  the  formulae  given  in  the 
following,  and  be  then  subtracted  from  the  entire  approaching 
water  Q,  for  which  case  Eq.  (19)  will  take  the  form 


Q  -C'-l 


In  the  equations  for  5X  and  S,  the  value  of  v  =  p  /L  is 

B  (k  +  H ) 

true  only  when  the  quantity  Q  arrives  immediately  in  front  of  the 
weir  with  a  velocity  equal  to  v. 

The  hydrodynamic  pressures  against  weir  surfaces  and  wing 
walls  previously  found  may,  however,  be  modified  for  the  waste 
water,  through  a  lateral  channel  or  submerged  weir,  when  a  definite 
disposition  has  been  decided  upon. 

That  the  above  formulae  may  be  generally  applied  to  complete 
overfall  weirs,  of  whatever  kind,  will  now  be  shown. 


20 


HYDRAULICS 


B.  Derivation  of  New  Formula  for  the  Discharge  over  Complete 
Over  jail  Weirs,  built  obliquely  to  a  Channel  or  Represented  in  Plan 
by  a  Curved  or  Broken  Line. 

When  it  becomes  desirable  to  prevent  excessive  rise  during  high 
water  stages,  or  when,  during  low  water  stages,  the  entire  flow  is 
to  be  utilized  through  a  lateral  flume,  this  may  best  be  accomplished 
by  a  diagonal  weir  of  sufficient  length. 

The  case  to  be  treated  is_shown  in  Fig.  9,  in  which  A  ADD 
represents  the  channel  and  EF  the  diagonal  weir,  which  is  also 
inclined  vertically  by  an  angle  ^r  with  the  horizontal.  The  nota- 
tion in  the  previous  article  will  be  retained. 


Fig.  9 


The  water  over  the  weir  crest,  having  a  depth  H,  quantity  BHv, 
and  a  velocity  of  approach  v  exerts  an  hydrodynamic  pressure 
against  the  weir  section  equal  to 

P  =    BH~. 


This  pressure,  being  the  same  in  all  points  of  the  section,  may 
be  considered  as  acting  in  the  axis  Oe  and  represented  in  magni- 
tude by  the  length  ae  =  P. 

Since  P  acts  on  the  weir  under  the_  angle  <£,  the  former  may  be 
resolved  into  components  eg  and  ef,  respectively  perpendicular 


COMPLETE  OVERFALL  WEIRS  21 

and  parallel  to  the  weir.     The  component  eg  then  affects  the 
direct  flow  over  the  weir  and  is  found  from  the  following  equations; 

eg  =  ae  sin  <£  =  yB  -  sin  $  . 
2  g 

If  this  resultant  normal  pressure  on  the  weir  section  be  repre- 

.  _  T> 

sented  by  a  rectangular  prism  of  length  EF  =  —  —  -  ,  height  H, 

and  thickness  /,  then  yBH  —  sin  c&  =  7  -  —  ,  from  which 

2g  sin  <j> 

t  =  —  sin2  (/>  =  S. 
2  g 

This  hydraulic  pressure  /  is  applied  at  the  surface  in  front  of  the 
weir  section  and  may,  therefore,  be  taken  equal  to  S,  as  was  done 
in  the  derivation  of  Eq.  (20). 

Taking  the  section  of  the  weir  as  in  Fig.  7,  and  considering  this 
section  normal  to  the  direction  of  the  weir  instead  of  parallel  to 
the  axis  of  the_  channel  as  before,  then  the  rectangle  EOE^R  (in 
which  now  EO  =  S)  will  represent  the  rectangular  prism  just 
mentioned.  _ 

Since  no  wing  walls  were  assumed  in  Fig.  9,  the  rectangle  OO^RR^ 
in  Fig.  7  becomes  superfluous. 

The  hydrostatic  pressure  of  the  advancing  water  on  the  weir  EF 
is  always  normal  to  the  weir  and  is 


This  pressure  may  again  be  represented  by  a  triangular  prism, 
as  was  done  in  the  derivation  of  Eq.  (20)  by  the  right  angled  equi- 
lateral triangle  O^s  in  Fig.  7. 

According  to  the  previous  derivation  of  Eq.  (20),  the  total 
hydrodynamic  pressure  against  the  weir  between  base  and  crest 

is  P5  =  7  —  Bk,  which  pressure  may  be  assumed  as  acting  along 

o  _ 

the  gravity  axis  Oe,  Fig.  9,  and  is  represented  by  the  line  a^. 


22  HYDRAULICS 

This  pressure  is  again  resolved  into  components  e^j^  and  e^  and 
from  the  parallelogram  of  forces  is  found : 

e.g.  =  die*  sin  <j>  =  7  —  Bk  sin  <b. 
g 

This  corresponds  to  the  force  o2o3  in  Fig.  5. 

That  portion  of  the  force  e^  which  acts  against  the  discharge 

area  in   the  direction  —   with  the  horizontal,  is  found  from  the 
2 

parallelogram  of  forces  and  is 
ozu  =  ujv=  0203  cos 

In  like  manner  is  found  the  horizontal  normal  hydrodynamic 
pressure  W{UD  from  the  parallelogram  rlwulwl1  Fig.  5,  as 

<\k  <y2  ^< 

w.w  =  u^w  cos  *  =  7  —  Bk  sin  <f>  cos2  —  . 
2          g  2 

Since  this  pressure  is  a  maximum  at  w  (the  crest  of  the  weir), 
and  diminishes  toward  the  water's  surface,  it  may  be  represented 

•n 

by  a  triangular  prism  of  the  length  — — -  ,  height  H,  and  bottom 
width  /?  found  from  the  equation, 


_____       ______  '/> 

OM  =  u.w=  0,0,  cos  —  =  7  —  Bk  sin  6  cos  — 

2  g  2 


w.w  =  7  —  Bk  sin  0  cos2  —  =  7  —  r-* 

2          2  sin 


n 

as  ?  =  —  =:  sin     >  cos   -    . 

2 


If  this  width  /?  be  applied  in  Fig.  7  as  52,  then  the  triangle  O^sz 
will  represent  a  section  of  the  prism  of  water  in  question. 

Accordingly  the  total  hydrodynamic  and  hydrostatic  pressures 
acting  at  the  level  of  the  weir  crest,  will  be  from  Fig.  7, 

-       —       —  2  tfk  & 

S=ER  +  Rs  +  sz  =  S  +  H  +  2-       sin2     cos2  i. 


Having  thus  found  the  hydraulic  pressures  both  at  the  water's 
surface  and  at  the  crest  of  the  weir,  the  quantity  of  flow  per  second 


COMPLETE  OVERFALL  WEIRS 


over  a  diagonal  weir  is  found  by  the  process  adopted  in  the  deriva- 
tion of  Eq.  (20)  as  follows: 


(25) 


When  it  is  desirable  to  construct  a  weir  such  that  the  maximum 
flow  shall  not  exceed  a  certain  assigned  limit,  that  the  weir  may 
become  safer  and  that  the  flow  be  more  or  less  deflected  away  from 
the  sides  of  the  canal  and  towards  its  axis;  or  when  it  is  necessary 
to  supply  lateral  channels  with  equal  quantities  of  flow  during  low 
and  average  stages  of  water,  one  of  the  following  modifications 
may  be  adopted : 

1.  The  weir  consists  oj  two  diagonal  parts  meeting  in  the  axis 
of  the  channel  and  forming  an  angle  with  the  apex  pointing  up- 
stream as  in  Fig.  10. 


Fig.  10 


Fig.  ii 


Fig.  12 


Fig-  13 


2.  The  weir  is  like  the  preceding  but  the  two  diagonal  sections 
are  separated  by  a  central  portion  EF  normal  to  the  direction  of 
flow,  as  in  Fig.  n. 


24  HYDRAULICS 

3.  The  weir  is  curved  to  the  arc  of  a  circle  with  convex  surface 
upstream  as  shown  in  Fig.  12. 

The  formulae  for  the  computation  of  flow  over  such  weirs  as 
shown  in  Figs.  10,  n,  and  12,  may  be  derived  from  equations  (20) 
and  (25). 

When  the  obliquity  of  the  two  halves  of  the  weir,  Fig.  10,  is  the 
same,  Eq.  (25)  will  apply  without  modification. 

For  weirs  in  plan  like  Figs,  n  and  12,  without  wing_walls,  the 
total  flow  is  found  by  considering  separately  the  parts  AE,  FD  and 
EF,  in  which  the  curved  portion,  EF,  Fig.  12,  is  considered  straight 
and  normal  to  the  direction  of  flow. 

The  flow_is_thus  separated  into  three  parallel  filaments,  the 
central  one  EF  exerting  a  normal  hydraulic  pressure  on  the  central 
area  of  flow,  found  from  Eq.  (22)  as 

/i-S'8]   .     .     (26a) 


in  which  S/  and  S'  have  the  following  values: 

The  pressure  of  the  flow  approaching  with  a  velocity  v  against 

v2 
the  weir  section  is  according  to  the  previous  derivation  p2  =  ybH  — 

hence  its  linear  magnitude  at  the  surface  per  unit  of  width  of  weir 


s 


as  represented  in  Fig.  7  by  the  line  EO  =  E^R. 

The  pressure  OOlt  resulting  from  the  wing  walls,  disappears  in 
the  present  case  because  no  walls  were  assumed. 

Likewise  the  normal  pressure  of  the  approaching  flow  against 
the  area  of  the  weir  below  its  crest  was  found  to  be 

7/2  <\fr 

P6=7  1L  wfcop^JE, 

g  2 

by  substituting  b  for  B,  because  in  the  present  case  the  width  b 
only  is  considered. 


COMPLETE  OVERFALL  WEIRS  25 

The  bottom  width  of  the  triangular  prism  representing  this 
pressure  was  previously  found  as 

2  v2Bk       ,  -»> 

P   =      7,     rr      COS     — 

bgH  2 

which,  by  substitution  of  b  for  B,  becomes 


The  magnitude  of  the  total  hydraulic  pressure  of  the  approach- 
ing water  at  the  level  of  the  weir  crest  is  found  from  Fig.  7  by  the 
equation 


2  k  (        Q 


or 


The  quantity  of  flow  per  second  over  the  middle  section  EF, 
Figs,  ii  and  12,  may  then  be  found  from  Eqs.  (26),  a,  b  and  c. 

The  flow  over  the  oblique  portions  of  the  weir  may  now  be  found 
from  Eq.  (25)  in  the  following  manner: 

Since  for  an  oblique  weir  the  hydraulic  pressure  of  the  advancing 
flow  at  the  surface  of  the  water  is  equal  to  S  and  at  the  crest  of  the 
weir  is  equal  to  S±  from  Eqs.  (25),  in  which  S  and  St  are  found  per 
unit  of  width  of  channel,  it  follows  that  these_equations  are  also 
applicable  to  the  oblique  weir  parts  AE  and  FD,  Figs,  n  and  12. 

Also,  the  length  of  each  oblique  weir  =  —  ;  —  -  and  for  the  two 

2  sin  <f> 

•n  _  r  T> 

=  -:  —    •     Then  by  substituting  for  -  —  -  in  Eq.  (25),  the  value 
sin  9  sin  9 

T)  1 

—  -  —  —  ,  the  quantity  of  flow  over  the  oblique  weir  sections  is 
sm  9 

found  to  be 


26  HYDRAULICS 

Hence,  the  total  flow  over  a  weir,  polygonal  or  circular  in  plan, 
as  in  Figs,  n  or  12,  is  equal  to 

Q  =  <2i  +  <22     ......    (26*) 

When  it  is  required  to  construct  a  weir  for  a  medium  stage  of 
water  such  that  the  high  water  mark  shall  not  be  materially  raised, 
the  disposition  shown_in  Fig.  13  has  of  late  been  applied,  by  making 
the  oblique  portion  OP  of  sufficient  length.  The  section  NO  =  b 
is  so  chosen  as  to_provide  for  the  partial  flow  of  the  mass  AN  Of) 
and  the  sections  OP  and  PR  are  so  dimensioned  as  to  admit  of  a 
regular  flow  of  the  quantity  O^OUR. 

By  giving  the  crest  of  the  weir  section  OP  a  slope  equal  to  the 
surface  slope  above  the  weir,  and  placing  the  weir  section  PR  level 
with  the  lower  point  P,  the  approaching  flow  will  reach  all  parts  of 
the  weir  with  the  same  initial  velocity  and  constant  depth  H. 

To  comply  with  the  conditions  of  uniform  flow,  it  is  necessary 
to  first  find  the  formulae  representing  the  flow  per  jsecond  over  each 
section  of  the  weir.  It  is  assumed  that  NO  =  PR  =  b,  and  that 
the  cross-section  NU  is  regular. 

Since,  for  the  case  in  hand,  <£  =  ^  =  90°  for  the  sections  NO 
and  PR,  the  formula  (22)  will  apply  to  these  sections  and  the  flow 
over  each  will  be  represented  by  the  equation 


q  -    .  pi,         g    --    [S,  -  S]   .     .     .     (27*) 

3  VJj  —  O/ 

Regarding  the  flow  over  the  section  OP,  it  must  be  considered 
that  since  the  water  flows  almost  parallel  to  this  section  with  a 
velocity  v,  the  hydrostatic  pressure  H  on  this  portion  of  the  normal 

0.67  v2 
section  would  be  diminished  by  an  amount  =  -     —  • 

2£     _         _ 

But,  since  the  water  flowing  between  cross-sections  OU  and  PR 
also  flows  over  the  weir  OP  diagonally  and  hence  exerts  an  hydro- 
dynamic  pressure  against  the  discharge  area  which  is  somewhat 

less  than  —  for  normal  impact,  it  would  seem  probable  that  these 


COMPLETE  OVERFALL  WEIRS  27 

two  opposite  effects  of  the  advancing  water  would  neutralize  each 
other,  and  hence  it  is  fair  to  assume  that  the  advancing  flow  exerts 
only  an  hydrostatic  pressure  H,  and  the  flow  over  the  section  OP 
may  be  found  from  the  simple  formula 

(27b) 


The  total  flow  over  the  weir  NOPR  will  then  be 

Q  =  2  g  +  fc (27c) 

Such  weirs  of  broken  lines  offer  the  advantage  that  the  back 
water  usually  caused  at  times  of  high  water  may  be  reduced  to  one 
half  that  which  would  result  from  a  weir  built  normally  to  a  stream, 
also  that  the  Opposite  bank  N  U  is  not  so  susceptible  to  washouts 
as  in  the  case  of  oblique  weirs. 


CHAPTER   III. 
INCOMPLETE   OVERFALL  WEIRS. 

Derivation  of  New  Formula  for  Discharge  over  Incomplete  Over- 
falls or  Submerged  Weirs,  and  through  Contracted  River  Channels. 

THE  same  rational  suppositions  employed  in  the  previous 
derivations  will  be  followed  in  the  present  chapter.  Accordingly, 
the  hydrodynamic  and  hydrostatic  pressures,  acting  on  the  dis- 
charge section,  will  be  ascertained  from  considerations  peculiar  to 
the  problem.  These  pressures  will  then  be  graphically  combined 
to  produce  the  resultant  pressure  areas  on  the  discharge  section, 
and  from  the  latter,  analytical  formulae  will  then  be  developed. 


Wingwall 


p 

*          i 

b                f     , 

V/L             M 

D 

Wing,|f 

j 
rjfc 

I 

D 

Fig.  14* 

Fig.  140  represents  the  longitudinal  section  of  a  river  or  canal,  ob- 
structed by  a  submerged  weir  and  wing  walls  A'E  and  D'F,  shown 
in  plan  Fig.  14^.  All  dimensions  are  indicated  on  the  two  figures. 

28 


INCOMPLETE  OVERFALL  WEIRS  29 

The  mean  depth  of  the  lower  pool  above  the  weir  crest  is 
Hl  =  Tl  —  k,  when  the  damming  effect  of  the  velocity  on  the 
lower  level  is  neglected. 

Let  Q  be  the  quantity,  per  second,  approaching  the  weir  with 
a  velocity  v,  which  after  being  discharged  over  the  weir  proceeds 
with  a  mean  velocity  F,  the  amount  of  which  velocity  depending 
on  the  nature,  form  and  slope  of  the  bed.  Also,  let  ^  be  the 
coefficient  of  flow  through  the  upper  part  section  EEit  Fig.  140, 
and  /^  for  the  lower  submerged  part  E^E2. 

The  flow  of  approach  exerts  an  hydrodynamic  pressure  against 
the  discharge  area  which  is  uniform  over  the  total  depth  H  and 
hence  may  be  represented  by  a  rectangular  prism  EOE2R,  of 


height  H,  length  b,  and  width  EO  =  Kfi  =  —  =  —  • 

2g        2g\BT/ 

The  hydrodynamic  pressure,  against  the  discharge  area,  result- 
ing from  the  flow  of  approach  against  the  two  wing  walls  and  over 
the  height  H,  is  again  represented  by  a  rectangular  prism  OJK.^RO 

of  length  b  and  width  Of>  =  R^R  =  r-f—  —  )  cos2^  • 

bg\    2    /        2 

The  hydrostatic  pressure  exerted  by  the  head  H2  against  the    v 
discharge  area  EE2  is  represented  by  the  area  O^S^,  being 
the  resultant  of  the  active  hydrostatic  pressure  O^T^R^  less  the 
back  pressure  EJL^e  from  the  quiescent  lower  pool.     However, 
the  lower  water  is  moving  away  from  the  discharge  area  with  a 

V2       nV2 

velocity  F,  thus  creating  a  suction  =  0.67  —  =  -  on  the  -part 

2  g       2  g 

section  EJ£V  which  will  diminish  the  residual  effect  of  the  hydro- 
static counterpressure  E^E2e  by  an  amount  represented  by  the 
rectangle  E^E^a.  The  triangle  /o^thus  represents  the  remain- 
ing  hydrostatic  back  pressure  on  j^E^  while  the  smaller  triangle 
fE^a  =  s{a2s3  represents  the  residual  pressure  effect  on  the 
discharge  area  EJ1  due  to  the  suction.  Hence,  by  neglecting  this 
small  suction  triangle  /£xa,  the  net  area,  E^aJ  =  s^^ss^  then 
represents  the  resulting  total  suction  on  the  section  E{E2.  Thus 
the  figure  EO^szsE2  represents  the  total  effective  pressure  on  the 


30  HYDRAULICS 

discharge  area  EE2,  except  that  due  to  impact  against  the  weir 


wallG. 


The  hydrodynamic  pressure  against  the  discharge  area  resulting 
from  the  velocity  of  approach  impinging  on  the  weir  wall  E2G 
was  found  by  Eq.  (15)  as 


g 


and  is  represented  by  the  line  w^w.  Fig.  5.  This  pressure  has  a 
maximum  effect  on  the  strata  along  sE2  and  may  be  regarded  as 
entirely  dissipated  at  the  level  /^g,  hence  the  triangle  sTs3  of 


_        /  nyz  \ 

height  /1£2  =  (ri-^  --  ]   and  base  ft  may    be    taken    to 

represent  this  pressure  PQ.    The  value  of  ft  is  then  found  from 


and  hence,     ft  = 


The  area  EOls3TE2  then  represents  the  total  effective  pressure 
on  the  discharge  area,  and  the  unit  pressures  along  certain  filaments 
of  the  discharge  may  now  be  determined. 

The  pressure  along  the  surface  element  O^E  =  S  becomes 


g 


The  filament  f^^  down   to  which  there  will   be   no  counter- 
pressure  from  the  lower  pool,  will  represent  a  pressure  5t  evaluated 

-f7I=5  +  tf2+^L          (,8ft) 


INCOMPLETE  OVERFALL  WEIRS  31 

Finally  the  pressure  along  the  filament  E2T,  at  the  crest  of  the 
weir,  is  found  as 

2  i?Bk  cos2  * 


The  flow  through  the  part  section  Ej^  proceeds  without  counter- 
pressure  as  for  a  complete  overfall  into  free  air,  and  by  observing 

nV2 
that  H2  H =  5j  —  S,  this  discharge  Ql  becomes,  according 

nV2 
to  Eq.  (19),  when  H2  H is  substituted  for  H, 


/2g[5^_55] (2&/) 

The  discharge  Q2  through  the  lower  portion  of  the  section  over 
the  height  f^E2  =  (Tl  —  k Jis  found  by  integration  as  was 

done  in  the  derivation  of  Eq.  (19).  The  velocity  along  /^3  will 
now  be  \/2  gSlt  and  that  along  E2T  will  be  \/2  gS2.  Q2  then 
becomes 

M-*-^ 

Q2  =  -  pj>  vx2  g  \ -2-£  I  (52S  -S?)   .     (28«) 

3  X  ^2~ ^1  ' 


If  the  immersed  height  j^E.^  is  small,  then  the  mean   velocity 

/      7^    +  5  \ 
through  this  portion  of  the  section  may  be  taken  asy  2  g  I— a  J, 

corresponding  to  a  pressure  height  — l—  — L ,  whence 


.     (28/) 
From  these  values  the  total  discharge  is  thus  found  to  be 


HYDRAULICS 


Equations  (28)  may  then  be  regarded  as  the  fundamental  equa- 
tions for  incomplete  overfalls  and  submerged  weirs.  By  introduc- 
ing special  values  as  was  done  in  Eqs.  (19)  and  (20),  the  following 
simpler  forms  are  obtained  : 

When  v  =  o,  then 


S  = 


2  g 


t  +  Q 


>.      .    (29a) 


wherein  H^  =  Tl  —  k. 

For  the  discharge  through  a  lateral  orifice  in  the  vertical  'wall  oj  a 
reservoir,  into  a  reservoir  oj  lower  level,  so  as  to  produce  an  incom- 
plete overfall,  and  assuming  quiet  water  in  each  reservoir,  then 
<£  =  ty  =  90°  and  v  =  V  =  o  and  calling  the  height  in  the  lower 
pool  above  the  bottom  of  the  opening  Hv  then  Eqs.  (28)  become 

S    =  o]  Sl  =  S2  =  H2\ 


This  formula  coincides  perfectly  with  Dubuat's  formula,  thus 
proving  that  the  latter  applies  only  for  the  special  case  here 
assumed. 

Figs.  15  and  i6a  represent  a  river,  the  flow  through  which  is 
obstructed  by  two  piers  A^E  and  DJ?  such  that  the  natural  surface 
PN  is  raised  to  the  level  EA.  The  pressures  active  on  the 
discharge  area  for  this  case  will  now  be  determined. 

The  hydrodynamic   pressure,   resulting   from   the   velocity  of 


INCOMPLETE  OVERFALL  WEIRS 


33 


approach  v,  is  uniformly  distributed  over  the  entire  depth,  and  is 
represented  by  a  rectangle  EORE2,  in  which  EO  =  —  . 


B    ^         22 

/•) 
/ 

Wlei, 

x^^^^                              N 

1 

/ 

IF 

[__,/ 

i 

\ 
\s 

R:R 


Fig-  I5 

The  hydrodynamic  pressure,  due  to  velocity  of  approach  and 
deflected  into  the  discharge  area  by  the  piers,  is  represented  by  the    v 
rectangle  OOfi^R,  the  width  of  which  is 


The  suction  caused  by  the  velocity  of  discharge  V  is  represented 
by  the  rectangle  E^E^a^j  which  subtracted  from  the  hydrostatic 
pressure  due  to  Tl  and  represented  by  the  triangle  Ef^e^  leaves 


A        \  \ 


*jt 

p      /A 


PI         4  D 


Fig. 


Fig. 


(after  neglecting  the  small  triangle  aEJ),  the  net  hydrostatic 
pressure  area  fa^e  =  s3Ts.  The  total  hydrostatic  pressure  result- 
ing from  the  head  T  is  given  by  the  triangle  OJR-^T  from  which 
the  net  counterpressure  s3Ts  is  subtracted  to  leave  the  effective 
pressure  area  O^^R^. 


34  HYDRAULICS 

The  total  pressure  on  the  discharge  area  is  thus  represented  by 
the  figure  EOls3sE2  from  which  unit  pressures  at  any  point  of  the 
discharge  section  may  be  obtained. 

The  discharge,  through  the  part  section  Eflt  takes  place  as  for 
discharge  into  open  air,  while  that  through  the  lower  portion  f^E2 
flows  with  a  velocity  corresponding  to  the  pressure  head  sE2. 
Hence,  the  following  pressures  are  obtained: 


- 

2  g 


&  r 

— 
2l 


b 
and 


.....     (306) 


CnV2  \ 
H2  H  --  1  =5.  —S, 
2  g  / 

Eq.  (19)  again  furnishes  the  value  for  the  flow  through  Ej^  as 


—  nV2 

Also,  the  discharge  through  the  part  section  /1£2=  T1  --  » 

subjected  to  counterpressure  and  flowing  under  constant  head,  is 


Finally,  the  total  flow  through  the  contracted  section,  of  height 
T  =  EE2,  and  width  b  =  EF,  Fig.  i6a,  becomes 


For  the  case  of  a  river  obstructed  by  piers,  as  in  Fig.  i6&,  the 
above  formulae  may  be  employed  by  substituting  the  aggregate 
width  of  the  piers  for  the  width  B  —  b  of  the  former  wing  walls. 
This  is  on  the  supposition  that  the  piers  are  sufficiently  near  to 


INCOMPLETE  OVERFALL  WEIRS  35 

each  other  so  that  their  individual  damming  effects  would  result 
in  a  uniform  elevation  of  the  approaching  surface. 

However,  this  distinction  must  be  made.  In  the  case  of  wings, 
only  two  contractions  occur,  while  for  pier  obstructions  these 
contractions  are  repeated  in  each  opening,  thus  necessitating 
special  values  for  the  coefficients  /*  and  /^  according  to  the 
observations  of  Francis. 

In  closing  this  subject  a  few  remarks  are  here  added  relative 
to  the  effect  of  obstructions  in  rivers,  and  the  permanency  of 
river  beds. 

From  many  accurate  measurements  and  gagings,  it  is  seen  that 
the  velocity  is  maximum  at  about  one  third  the  depth  and  dimin- 
ishes towards  the  bottom  in  all  cases  of  unobstructed  uniform  flow. 
The  discharge  curve  is  then  of  parabolic  form  as  shown  by  E1SNV 
Fig.  146. 

On  the  other  hand,  for  obstructed  flow,  especially  as  in  Fig.  140, 
the  velocity  at  the  surface  is  always  diminished.  It  increases 
gradually  with  the  depth  down  to  the  point  jv  where  the  counter- 
pressure  from  the  lower  pool  becomes  active,  and  below  this  point 
it  remains  practically  constant.  This  is  illustrated  by  the  discharge 
curve  EfiE^  Fig.  146. 

This  observation  explains  the  cause  for  the  extensive  erosions 
which  always  follow  the  placing  of  a  pier  or  other  obstruction  into 
a  river  channel. 


CHAPTER   IV. 
SLUICE  WEIRS  AND  SLUICE  GATES. 

Derivation  of  New  Formula  for  Discharge  over  Sluice  Weirs  and 
through  Sluice  Gate  Openings. 

IN  deriving  formulae  for  discharge  through  sluice  openings,  it  is 
usually  supposed  that  the  efflux  takes  place  as  for  a  vessel,  Fig.  17, 
the  inside  surface  of  which  is  assumed  to  be  frictionless  and  gradu- 


Fig.  17 


Fig.  i 8 


ally  contracted  into  a  conical  shape,  so  that  the  velocity  V,  in  the 
enlarged  section  M 1M,  is  gradually  accelerated  toward  the  opening 
ATjAT.  The  water  in  falling,  through  the  height  H,  acquires  a 
certain  vis  viva,  none  of  which  is  supposed  to  be  lost  in  the  produc- 
tion of  acceleration.* 

Now  if  the  flow  through  such  a  vessel  be  compared  with  that  in  a 
river  contracted  by  sluice  weirs,  Fig.  18,  it  is  almost  self-evident 
that  this  condition  of  flow  in  no  wise  resembles  the  above  illustration 
in  Fig.  17. 

*  Ruehlmann,  Hydromechanik,  pp.  207,  208,  463  and  464. 
36 


SLUICE  WEIRS  AND  SLUICE  GATES 


37 


The  water  in  the  river  impinging  on  the  wing  walls,  projections, 
corners,  etc.,  produces  impact.  Also,  the  flow  does  not  arrive  at 
the  discharge  area  with  a  gradual  motion,  but  rather  suddenly, 
and  is  then  deflected  at  various  angles  toward  the  openings. 

All  this  goes  to  prove  that  a  considerable  portion  of  the  vis  viva 
stored  in  the  flow  of  approach,  is  necessarily  lost  by  impact  and 
friction. 

The  older  formulae  were  also  based  on  the  erroneous  supposition 
that  the  discharge  over  a  submerged  weir  was  resisted  by  an  hydro- 
static counterpressure  on  the  discharge  area  just  as  for  discharge 
into  quiet  water,  thus  disregarding  the  suction  due  to  velocity  of 
discharge. 

Formulae  based  on  such  discordant  ideas  cannot  be  expected  to 
furnish  correct  values  for  discharge  through  sluices,  etc.  For  this 
reason  new  and  more  rational  formulae  will  now  be  developed 
with  the  aid  of  the  principles  formerly  applied  to  the  derivation  of 
Eqs.  (19),  (20)  and  (28). 

1.  Formula  for  case  illustrated  in  Fig.  19,  being  a  sluice  weir 
with  submerged  discharge. 


Fig.  19 

Figures  18  and  19  represent  the  plan  and  longitudinal  section, 
respectively ,_of_  a  submerged  weir  AJ)V  built  in  a  river.  The 
wing  walls,  A,E  and  Df,  extend  above  water  level  and  the  space 
EF,  Fig.  1 8,  contains  three  gate  openings. 


38  HYDRAULICS 


When  the  gates  are  raised  to  a  height  E^E2  =  a,  the  water  in  the 
upper  pool  approaching  with  a  velocity  V,  will  be  supposed  to  exert 
a  pressure  head  Hv  causing  a  velocity  of  discharge  F,  in  the  lower 
pool. 

The  quantity  of  discharge  Q,  passing  through  the  three  openings, 
as  above  illustrated,  will  now  be  found. 

The  hydrodynamic  and  hydrostatic  pressures,  exerted  on  the 
discharge  area,  are  separately  determined  for  each  of  the  three 
following  prisms  of  flow : 


1.    The  lower  prism  CGKE2,  of  height  k. 


2.  The  middle  prism  KEJEtJt  of  height  a. 

3.  The  upper  prism  JEJEA,   of  height 


Making  the  sum  of  the  widths  of  the  three  gates  equal  to  £>,  then 
the  total  width  of  the  interposed  obstruction,  including  piers  and 
wing  walls,  will  be  B  —  b. 

The  hydrodynamic  pressure  against  each  of  the  orifices  may  now 
be  represented  by  a  rectangular  prism  of  height  a,  length  B  —  b, 

v2 
and  width  —  ,  shown  in  Fig.  19,  as  pressure  area  I. 

2  g 

The  hydrodynamic  pressure  deflected  by  the  wing  walls  and  piers 
into  the  discharge  area,  as  found  in  the  derivation  of  Eqs.  (19) 
and  (20),  may  be  represented  by  a  rectangular  prism  II,  of  width 


/  = 


However,  the  wing  walls  in  the  present  case  being  perpendicular 
to  the  canal  axis,  <£  =  90°  and  hence, 


SLUICE  WEIRS  AND  SLUICE  GATES  39 

The  upper  prism  of  flow,  of  depth  H  —  a,  exerts  an  hydro- 
dynamic  pressure  against  the  three  gates_and  the  upper  part  of  the 
wing  walls,  active  along  the  gravity  axis  de  and  over  an  area 


This  pressure  is  then 

H2-(k  +  a)].     .     .    .     (32) 


Since  the  lower  strata  /£xof  this  prism  undergo  scarcely  any 
deflection,  while  the  upper  strata  along  AE  must  be  deflected  90°, 
the  average  deflection  for  the  prism  may  be  taken  as  45°.  This  is 
represented  in  Fig.  19,  by  the  direction  EJ.  _ 

If  the  pressure_j>  be  represented  as  a  linear  magnitude  d^elt 
decomposed  into  ej  and  e^g,  then, 

^/  =~d^  cos  45°  =  yB  [rt+  H2-(k  +  a)]  ~-  cos  45°. 

But  ej  =  E^e^  may  be  again  resolved  into  components  E^e  and 
EjT  ,  from  which  the  required  horizontal  component  is  found  as 


£     =  E      cos  45°  =  VB  [7\  +H,  -  (k  +  a)]  -       .     .   (33) 

o 

This  component  exerts  a  uniform  pressure  on  the  discharge  area 
and  its  effect  is  represented  by  a  rectangular  prism  of  length  b, 
height  a,  and_width  tv  which  latter  may  be  found  from  Eq.  (33) 
by  making  £/  =  yabtv  whence 


which  determines  the  pressure  area  III,  Fig.  19. 

The  total  hydrostatic  pressure  of  the  upper  pool  on  the  whole 
discharge  area  may  be  represented  by  the  weight  of  a  triangular 
prism  o^T.  The  counterpressure  from  the  lower  pool  is  then 
given  by  the  triangular  prism  UJ  =  i2sT,  which,  subtracted  from 


40  HYDRAULICS 

the  former,  gives  the  net  effective  hydrostatic  pressure,  represented 
by  the  area  o^R^. 

The  portion  o^p2s2i2y  of  this  latter  area,  is  taken  up  by  the  upper 
portion  of  the  sluice  gate,  leaving  only  the  effective  pressure  area 
o2R^ss2  =  vbH2a,  shown  in  Fig.  19,  as  area  IV  of  width  R^s  =  H  2. 

The  discharge  velocity  F,  in  the  lower  pool,  produces  a  suction 


-  ,  represented  by  the  pressure  area  V,  of  width 


The  triangular  prism  VI,  with  base  T^3  =  /?,  represents  that 
part  of  the  hydrostatic  pressure  on  the  weir  E2G,  which,  after  being 
twice  deflected,  finally  becomes  effective  on  the  discharge  area. 
In  the  derivation  of  Eqs.  (19)  and  (20),  the  value  of  /?  was  found  as 


. 
abg  2 

The  total  hydraulic  pressure  of  the  upper  pool  on  the  discharge 
area,  is  then  obtained  from  the  summation  of  all  the  pressure 
areas  above  found.  This  gives  the  trapezoidal  figure 


ElSlT,E2  =  I  +  II  +  III  +  IV  +  V  +  VI  .     .    (35) 

from  which  the  unit  pressure  on  any  point  of  the  discharge  section 
may  be  ascertained. 

Thus,  the  pressure  in  the  upper  filament  s1El  —  S  is  made  up 
of  the  combined  widths  of  the  several  prisms  I  to  V,  and  is  evalu- 
ated as 


2g 
or 


=I++ 

2  g\_          2b       2  ab\  /J  2  g 

The  unit  pressure  along  the  filament  T1E2  =  Sl  will  then  be 


abg 


SLUICE  WEIRS  AND  SLUICE  GATES  41 

By  a  process  similar  to  that  employed  in  deriving  Eq.  (19),  the 
quantity  of  discharge  is  found  to  be 


When  the  sluice  sill  is  level  with  the  river  bed,  making  k  =  o,  and 
thus  Sj  =  5,  then  the  flow  through  a  proceeds  with  the  uniform 
velocity  \/  2  gS,  and  the  discharge  may  be  found  from  the  following 
simple  equation  : 

Q  =  ^ 


2.  Formula  for  the  case  illustrated  in  Fig.  20,  for  k  =  o  and 
Tl  <  a,  will  now  be  derived. 

The  rectangles  I,  II,  and  III  represent,  as  in  Fig.  19,  the  cross 
sections  of  the  pressure  prisms  effective  on  the  discharge  area  and 
resulting  from  the  hydrodynamic  pressure  due  to  velocity  of 


Fig.  20 

approach  impinging  on  the  discharge  area  and  on  the  wing  walls 
and  upper  portion  of  the  gate.  Hence,  the  expressions  just  found 
will  apply  to  the  present  case  when  k  =  o  is  introduced. 

Considering  the  hydrostatic  counterpressure  of  the  lower  pool 
PN,  active  up  to  a  line  sjnv  then  the  effective  pressure  on  the 
discharge  area  will  be  given  by  the  triangle  oJK.^T  less  the  triangle 
o^oJK.  less  the  triangle  i2sT,  leaving  the  net  area  o2Rtsiv  which  is 
composed  of  areas  IV  and  V,  Fig.  20. 


HYDRAULICS 


The  area  VI  finally  represents  the  suction  on  the  discharge  sec- 
tion produced  by  the  discharge  velocity  F,  and  this  suction  being 
taken  as  uniformly  distributed,  its  horizontal  intensity  becomes 

nV2 
S^~     2g' 

Fig.  20  then  furnishes  the  unit  pressures  along  any  horizontal 
filament. 

From  this  figure  also,  the  following  dimensions  are  found: 


Ef,  =  o^  =  o2K  =  H2  +  7\  -  a 


2g 


nV2 


E2m 


T     _ 
~~ 


(37) 


The  discharge  is  evaluated  in  two  partial  quantities,  Qifor  the 
flow  through  the  upper  part  section  E^,  which  takes  place  as  for 
discharge  into  free  air,  and  Q2  for  the  flow  through  the  lower  part 
section  E2m,  which  is  submerged. 

Accordingly  the  resultant  hydraulic  pressure  along  the  filament 
li^K  =  5,  is  found  as 


or 


g 


The  total  pressure  in  the  lower  filament  5tw,  is  equal  to  S  4-  Ejn 


=  S19  or 


S,= 


—  -Tl.    .     .    (386) 

2g 


SLUICE  WEIRS  AND  SLUICE   GATES 


43 


The  discharge  Qlt  according  to  the  fundamental  Eq.  (19),  now 

becomes 

nV2 


Ql= 


a  + 


-r, 


The  resultant  hydraulic  pressure  over  the  lower  part  section  E2m 
is  uniformly  distributed  over  this  depth  and  has  the  unit  intensity 
St,  corresponding  to  a  velocity  \/2  gSv  from  which  Q2  becomes 


=  ,6  ( 


r  - 


The  total  discharge  through  the  orifice  of  height  a  is  then 

Q  -  Q,  +  C2  .......    (38*) 

3.    Another  case  of  discharge  through  sluices  is  illustrated  in 
Fig.  21.     Here  the  hydraulic  pressure  is  supposed  to  be  very  great, 


Fig.  21 

and  the  discharge  area  very  small.  The  velocity  at  the  orifice  is 
greater  than  the  discharge  velocity  and,  in  consequence  of  this 
retardation,  an  impact  is  produced  which  is  expended  in  elevating 
the  discharge  surface  into  a  wave  just  in  front  of  the  sluice  opening. 

Regarding  this  phenomena  and  its  effect  on  the  discharge,  little 
reliable  information  exists.  The  following  solution  is  offered. 

In  the  previous  case  all  the  factors  bearing  on  the  discharge  were 


44  HYDRAULICS 

carefully  considered  and  these  apply  equally  in  the  present  problem 
with  the  exception  that  the  wave,  when  it  exists,  enters  as  a  disturb- 
ing element  in  determining  the  particular  value  of  Tv  which  is 
likely  to  govern  the  discharge.  The  matter  then  resolves  itself 
into  finding  such  a  suitable  value  for  Tr 

Obviously,  the  whole  height  of  the  wave  cannot  be  taken  as  the 
level  of  the  lower  pool,  because  the  wave  is  merely  the  result  of 
impact  of  discharge,  and  because  the  wave-crest  is  immediately 
followed  by  a  deep  wave-hollow. 

Also,  the  velocity  at  different  points  of  the  wave-crest  varies  in 
magnitude  and  direction,  and  no  particular  value  could  be  selected 
as  the  discharge  velocity. 

When  the  normal  flow  is  once  established,  the  depth  and  velocity 
in  the  lower  pool  undoubtedly  exert  a  marked  influence  on  the 
production  and  shape  of  the  wave,  which  effect  is  probably  trans- 
mitted back  to  the  discharge  area. 

Slight  changes  in  the  conditions  of  flow,  constantly  occurring, 
may  cause  the  wave  to  disappear  entirely,  thus  reducing  the  surface 
of  the  lower  pool  to  a  level. 

It  would  thus  seem  most  rational  to  disregard  the  wave  and 
assume  as  the  most  probable  lower  pool  level,  the  one  resulting 
from  normal  flow. 

However,  should  the  length  of  the  flume  be  insufficient  to  develop 
a  continuity  of  flow  through  the  flume,  then  the  mean  height  of  the 
wave  may  be  taken  as  the  resisting  depth  of  the  lower  pool,  and 
the  mean  velocity  resulting  from  this  depth  may  then  be  used  in 
the  above  formulae. 

The  values  of  Tl  and  F,  in  the  formulae  of  the  present  chapter, 
are  supposed  to  be  known  from  gaugings  or  otherwise.  However, 
when  they  are  not  known,  as  in  the  case  of  proposed  sluices,  then 
the  methods  suggested  in  Chapter  VI  must  be  employed. 

In  the  previous  formulae,  a  certain  velocity  of  approach  v  was 
included.  When  discharge  takes  place  from  a  lake  or  other 
quiescent  water,  then  v  =  o,  and  the  various  pressure  areas  I,  II, 
and  III  disappear,  thus  greatly  simplifying  the  formulae. 


SLUICE  WEIRS  AND   SLUICE  GATES 


45 


This  special  condition  when  introduced  into  the  above  equations 
furnishes  the  following  formulae  for  discharge  from  quiescent 
water  into  a  flume  or  canal. 

a.  For  a  sluice  gate  at  the  inlet  to  a  canal  supplied  from  a  lake, 
case  Fig.  19,  then  v  =  o  and  Eqs.  (36)  give 


=  5, 


—   ^ 


•    •    (39) 


b.    For  a  sluice  gate  at  the  inlet  to  a  canal  supplied  from  a  lake, 
case  Fig.  20,  then  v  =  o,  and  Eqs.  (38)  become 

S  =  H2+T1-a, 


.     .     .     .     (4o) 


c.  Should  the  water  be  discharged  from  a  high  reservoir  into 
one  of  lower  level,  by  means  of  a  sluice  gate,  both  reservoirs  being 
quiescent  and  the  lower  level  is  above  the  bottom  of  the  sluice 

nV2 
opening,  then  the  function  =o,  and  the  hydrostatic  counter- 

2  g 
pressure  from  the  lower  reservoir  then  becomes  effective. 

When  the  discharge  of  a  river  is  known,  then  the  dimensions  of 
weirs,  sluices,  etc.,  as  well  as  backwater  height  and  distance,  can 
be  determined  by  a  method  of  approximation  to  be  discussed  in 
Chapter  VI,  with  the  aid  of  the  above  formulae. 

By  the  application  of  -  the  principles  here  employed,  in  the 
derivation  of  new  formulae  in  Chapters  II,  III,  and  IV,  it  will  be 
possible  to  solve  any  similar  problems  by  developing  formulae  appro- 
priate to  such  cases. 

The  formulae  of  the  present  chapter  are  also  applicable  to  small 
regulating  gates. 


CHAPTER  V. 

BACKWATER   CONDITIONS. 

Discussion  and  Formula  for  Backwater  Height  and  Distance. 

THE  formulae  in  the  previous  chapters  deal  with  problems  of 
obstructed  flow  due  to  objects  of  assigned  dimensions,  and  the 
backwater  height  enters  as  a  given  function.  In  the  following, 
the  reverse  conditions  will  be  treated  under  three  cases. 

1.  When  the  dimensions  of  an  existing  or  proposed  weir  or 
sluice  are  given  together  with  the  flow  of  approach,  to  find  the 
backwater  height. 

2.  The  flow  of  approach  and  backwater  height  being  given  to 
find  the  dimensions  of  the  weir  or  sluice  such  that  the  flow  will 
proceed  with  this  assigned  height. 

3.  Given  the  backwater  height  to  find  the  backwater  distance 
and  surface  curve. 

When  the  flow  of  approach  Q,  is  given,  as  for  problems  under 
i  and  2,  then  one  of  the  unknowns  H2,  k  or  b,  can  be  found 
for  assigned  values  of  the  others,  by  using  one  of  the  above  for- 
mulae. When  these  formulae  become  too  complicated  for  direct 
solution  of  an  independent  variable,  then  the  method  of  approx- 
imation, by  substitution  of  assumed  values  for  this  variable,  must 
be  resorted  to. 

Problems  coming  under  the  third  head  may  be  solved  by  the 
somewhat  complicated  formulae  given  by  Professor  Ruehlmann, 
which  in  the  absence  of  a  better  solution  are  here  reproduced. 

Drift  and  sedimentation  always  enter  as  a  disturbing  element  in 
river  hydraulics,  so  that  no  formulae,  however  accurate,  could  be 
made  to  permanently  satisfy  all  these  changeable  conditions. 

46 


BACKWATER  CONDITIONS  47 

Professor  Ruehlmann  gives  tabulated  values  for  the  ready  solu- 
tion of  his  formulae,  and  the  results  so  obtained  agree  very  well 
with  those  of  Hagen,  Weisbach  and  Heinemann. 

Let  D  =  the  original  uniform  depth  of  the  river. 
s    =  the  original  natural  slope  of  the  river. 
Z  =  the  total  backwater  height  measured  above  the  natural 

surface  slope. 
L  =  the  total  backwater  distance  measured  from  the  crest 

of  the  weir. 
z   and  /  are  coordinates  of  the  backwater  curve,  referred 

to  a  point  O  on  the  natural  slope  line,  vertically 

above  the  crest  of  the  weir.     (See  Fig.  22.) 


; 
~  >j 


Fig.  22 


f  —  )  and  /  f  —  Jare  Ruehlmann's  functions  of  —  and  —respec- 


tively, the  values  for  which  are  given  in  Table  I. 
Ruehlmann's  formula  is 


When  D,  s,  Z  and  z  are  given,  the  problem  is  thus  solved.  The 
different  values  of  (TV)  and  I  —  J  are  found  from  one  of  the  columns 
i,  of  Table  I,  and  opposite  these  in  column  2,  are  the  corresponding 
functions  /f-jand  /nU«  These,  when  substituted  in  Eq.  (41), 

give  the  abscissa  /  corresponding  to  the  ordinate  z,  for  any  given 
D  and  s. 


48  HYDRAULICS 

When  Dj  s,  Z  and  /  are  given,  the  value  of  z  is  found  from 
Eq.  (41),  by  transformation,  thus 

Z\      si 

(42) 


Then  having  found  fl-J  from  Eq.  (42),  the  corresponding  value 

of  (j\ is  given  by  Table  I   finally  z  =  j-  .  D. 

In  the  same  manner  Z  may  be  found  from  the  following  Eq.  (43) : 
./Z\     si        f/z\ 

f$rD+J\f>) (43) 

For  the  total  backwater  distance  L  the  three  quantities  z}  (— j 
and  /  f  -  J  are  each  zero.  The  Eq.  (41)  then  becomes  : 

*-?/© <«> 

When  Table  I  does  not  include  exactly  the  values  for  any  given 
(  n)OIm)'  t*ien  f  ( n)or  ^  (n)'  corresponding  to  tne  exact  given 
value,  may  be  found  from  the  following  interpolation  formula 

/( —  )  or  /  ( ~~"  1  ==  'V  :^  'V   ~f"  ( 'V  —  "V  )  I  ^  I  (A  f  \ 

\  7~)/          V  7~)/  ^  I  /y   —  v   I  v"o/ 

in  which 

^c    =  the  given  value  of  f  —  J  or  f  —  J 

/  z  \ 
x,  =  the  next  smaller  value  of  x  or  [-  )  in  Table  I. 


x2  =  the  next  larger  value  of  x  or  f  —  J  in  Table  I, 

y    =  the  required  function  /rj>)  • 

fZ\ 
y1  =  the  value  /f-    corresponding  to  xr 


fZ\ 
y2  =  the  value  /(-J  corresponding  to  xv 


CONDITIONS 


49 


TABLE   I. 

PROFESSOR  RUEHLMANN'S  BACKWATER  FUNCTIONS  FROM  HYDROMECHANIK, 

PAGE  484. 


I 

z 

D 

z 
D 

'(1) 

•ft) 

i 
Z 
D 

z 
D 

•(I) 

•6) 

i 
Z 
D 
z 
D 

•(I) 
•ft) 

i 
Z 
D 

z 
D 

•si 
•ft) 

I 

z 

D 

z 
D 

'(i) 
•fe) 

O.OIO 

0.0067 

0-235 

1.2148 

0.460 

i  .  6032 

0.685 

1.9077 

0.910 

2.1800 

0.015 

0.1452 

0.240 

1.2254 

0.465 

i.  6106 

0.690 

1.9140 

0.915 

2.1858 

O.020 

0.2444 

o.  245 

•2358 

0.470 

.6179 

0.695 

1.920;; 

0.920 

2.1916 

O.025 

0.3222 

0.250 

.2461 

o-475 

.6252 

0.700 

1.9266 

0.925 

2.1974 

0.030 

0.3863 

o-255 

•2563 

0.480 

.6324 

0.705 

1.9329 

0.930 

2  .  2032 

0.035 

0.44II 

0.260 

.2664 

0.485 

.6396 

0.710 

1.9392 

o-935 

2  .  2090 

O.O40 

0.4889 

0.265 

.2763 

0.490 

.6468 

0.715 

1-9455 

0.940 

2.2148 

0.045 

0.5316 

0.270 

.2861 

0-495 

.6540 

0.720 

I-951? 

0.945 

2.22O6 

0.050 

0.5701 

0.275 

.2958 

0.500 

.6611 

0.725 

1-9579 

0.950 

2.2264 

0.055 

0.6053 

0.280 

•3054 

0.505 

.6682 

0.730 

1.9641 

0-955 

2.2322 

O.O60 

0.6376 

0.285 

•3149 

0.510 

-6753 

o-735 

I-9703 

0.960 

2  .  2380 

0.065 

0.6677 

0.290 

•3243 

0.515 

.6823 

0.740 

I-9765 

0.965 

2-2438 

O.070 

0.6958 

0.295 

•3336 

0.520 

.6893 

0-745 

1.9827 

0.970 

2.2496 

0.075 

O.7222 

0.300 

.3428 

0-525 

.6963 

0.750 

1.9888 

0-975 

2.2554 

0.080 

0.7482 

°-3°5 

•35i9 

0-53° 

.7032 

o-755 

1-9949 

0.980 

2.26ll 

0.085 

o.  7708 

0.310 

.3610 

o-535 

.7101 

0.760 

2.0010 

0.985 

2.2668 

0.090 

0.7933 

o^S 

.3700 

0.540 

.7170 

0.765 

2.0071 

0.990 

2.2725 

0.095 

0.8148 

0.320 

-3789 

o-545 

•7239 

0.770 

2.0132 

0-995 

2.2782 

O.  IOO 

0.8353 

0-325 

.3877 

0-55° 

.7308 

0-775 

2.0193 

.000 

2.2839 

0.105 

0.8550 

0.33° 

-3964 

o-555 

-7376 

0.780 

2.0254 

.  IOO 

2.3971 

O.IIO 

0.8739 

o-335 

.4050 

0.560 

.7444 

0.785 

2.0315 

.200 

2.5683 

0.115 

0.8922 

0.340 

.4136 

0-565 

•7512 

0.790 

2.0375 

.300 

2.6l79 

O.  I2O 

0.9098 

0-345 

.4221 

0.570 

.7589 

0-795 

2.0435 

.400 

2.7264 

0.125 

0.9269 

o.35o 

.4306 

0-575 

.7647 

0.800 

2.0495 

-5° 

2.8337 

0.130 

0.9434 

0-355 

-439° 

0.580 

.7714 

0.805 

2-0555 

.60 

2  .  9401 

0.135 

o-9595 

0.360 

•4473 

0-585 

.7781 

0.810 

2.0615 

.70 

3-0458 

o.  140 

o-975i 

°-365 

•4556 

0.590 

.7848 

0.815 

2.0675 

i.  80 

3.1508 

0.145 

0.9903 

0.370 

.4638 

o-595 

.7914 

0.820 

2.0735 

1.90 

3.2553 

0.150 

i  .  005  i 

o-375 

.4720 

0.600 

.7980 

0.825 

2.0795 

2.00 

3-3594 

0.155 

1.0195 

0.380 

.4801 

0.605 

.8046 

0.830 

2.0855 

2.  10 

3-463I 

o.  160 

1-0335 

0.385 

.4882 

0.610 

.8112 

0-835 

2.0915 

2.  2O 

3-5564 

0.165 

1.0473 

0.390 

.4962 

0.615 

.8178 

0.840 

2.0975 

2.30 

3.6694 

0.170 

i.  0608 

0-395 

.5041 

o.  620 

.8243 

0.845 

2.1035 

2.4O 

3.7720 

o.i75 

i  .  0740 

0.400 

•5H9 

0.625 

.8308 

0.850 

2.1095 

2.50 

3-8745 

0.180 

1.0869 

0.405 

•5J97 

0.630 

.8373 

0-855 

2.II54 

2.6o 

3.9768 

0.185 

1.0995 

0.410 

•5275 

o-635 

.8438 

0.860 

2.I2I3 

2.70 

4.0789 

0.190 

1.1119 

0.415 

•53S3 

0.640 

-8503 

0.865 

2.1272 

2.80 

4.1808 

0.195 

1.1241 

0.420 

•543° 

0.645 

-8567 

0.870 

2.I331 

2.90 

4.2826 

0.200 

1.1361 

0.425 

•55°7 

0.650 

.8631 

0.875 

2.1390 

3-o° 

4-3843 

0.2O5 

i.  1479 

0.430 

•5583 

0-655 

.8695 

0.880 

2  .  1449 

3-5° 

4.4891 

O.2IO 

I-.J595 

o-435 

5659 

0.660 

•8759 

0.885 

2.1508 

4.00 

5-3958 

0.215 

1.1709 

0.440 

5734 

0.665 

.8823 

0.890 

2.1567 

4-5° 

5.8993 

0.220 

1.1821 

o.445 

5809 

0.670 

.8887 

0.895 

2.1625 

5.00 

6.4I2O 

0.225 

1.1931 

0.450 

5884 

0-675 

.8951 

0.900 

2.1683 

0.230 

i  .  2040 

o-455 

5958 

0.680 

1.9014 

0.905 

2.1742 

The  above  table  is  applicable  for  any  units  of  measurement,  as  feet  or  meters. 


50  HYDRAULICS 

The  application  of  Table  I  and  formula  (41)  will  now  be  illus- 
trated by  citing  two  examples  given  by  Professor  Ruehlmann  in 
his  Hydromechanik. 

Example  I.  Given  a  river  80  feet  wide  and  4  feet  deep  with  a 
slope  s  =  0.000623.  A  weir  built  in  this  river  raises  the  water 
3  feet  at  the  weir  site.  At  what  distance  back  from  the  weir  will 
the  backwater  height  be  0.25  feet? 

Here  (—}  =  ^  =  0.75.     From  the   table,  find   the  value  for 

/  f — )   =  1.9888.     Also    (—)  =  —  - —  =  0.0625   for  which  no 
\DJ  \DJ        4X4 

value  can  be  found  in  column  i,  hence  interpolation  by  Eq.  (45) 
becomes  necessary.     Thus  : 

Cf 

for       ~-  =  xl  =  0.060,   column  2  gives  /(^)=  yl  =  0.6376, 
for       —  =  x2  =  0.065,   column  2  gives  /(#2)=  y2  =  0.6677. 

By  substituting  these  values  in  Eq.  (45)  the  value  of  /  /  •—  J  corre- 
sponding to  —  =  x  =  0.0625  is  found  as 

;     =0-6526. 

Having  the  value  of  /  r^rh  the  required  backwater  distance  is 
found  from  Eq.  (41)  as 

/  = 4_  [1.9888-0.6526]  =  8579  feet. 

0.000623 

Also  the  total  backwater  distance  as  given  by  Eq.  (44)  is 

L  = 4_  x  I<9888  =  12,770  feet. 

0.000623 

Example  II.  Given  two  points  A  and  B,  distant  2020  meters 
apart.  What  backwater  height  Z  must  be  produced  by  a  weir  at  A 


BACKWATER  CONDITIONS 


such  that  the  backwater  height  at  B  is  0.891  m.,  when  the  original 
depth  of  water  is  1.59  m.,  the  fall  in  the  river  bed  between  A 


Fig.  23 

and  B  is  1.737  m->  and  the  discharge  is  158.52  m.3  per  second. 
See  Fig.  23. 


Then  si 
si 


2020 


X  2020  =  1.737  and  the  depth  D=  1.59  m.; 


hence  -— =  1.0924  and  -1  =  — 


D 


D       1.59 
1.7444  and  from  Eq. 


o .  560.    Then  from  Table  I, 


=  I-°924+  1-7444 


2 . 8368  for  which  the  table  gives  very  nearly  —  =  i .  50,  hence 
2  =  1.50  X  1.59  =  2.385  m. 


CHAPTER   VI. 
FLOW  IN  RIVERS  AND  CANALS. 

Derivation  of  Formula  lor  Discharge  from  Rivers  or  Lakes  into 
Water  power  Canals  and  Flumes. 

1.    General  Discussion. 

THE  following  formulae  are  intended  to  serve  the  engineer  in 
designing  flumes  or  canals  for  manufacturing  or  water-power 
purposes.  It  is  proposed  to  discuss  fully  the  various  short- 
comings of  older  formulae,  and  to  show  the  general  applicability 
of  the  foregoing  principles  and  ideas  to  problems  relating  to 
cross-sections,  slopes  and  inlets  necessary  for  the  delivery  of  certain 
quantities  of  water  through  flumes  and  canals  which  are  supplied 
either  from  rivers  or  lakes. 

Ruehlmann,  Hydromechanik,  p.  439-442,  says,  for  the  case  of 
such  canal  inlets  without  regulating  works,  "  from  a  scientific 
standpoint,  the  presentation  of  this  problem  still  lacks  all  mathe- 
matical demonstration,  and  for  practical  purposes  we  must  con- 
tent ourselves  with  a  few  observations  made  by  Dubuat."  Based 
on  these  observations  Ruehlmann  then  offers  the  following  doctrine : 


Fig.  24 

"  The  mean  velocity  and  area  of  flow  in  any  canal  or  flume  of 
constant  width  and  uniform  slope,  are  so  related  that  velocity 

52 


FLOW  IN  RIVERS  AND   CANALS  53 

height  is  equal  to  the  difference  in  level  between  the  supply  reser- 
voir at  the  inlet  and  the  water  in  the  flume  at  a  point  where  the 
velocity  of  discharge  has  just  become  uniform." 

Referring  to  Fig.  24  (which  is  Ruehlmann's  Fig.  167)  and 
calling  V  the  mean  velocity  of  discharge  through  the  section 
AC,  v  the  velocity  of  approach,  m  a  coefficient  of  contraction  and 
other  dimensions  as  indicated  on  the  figure,  then,  according  to 
Ruehlmann, 

i.  V2      v2 

e  —  el  = -  —  —  ' (46) 

2  gm2     2  g 

When  v  is  very  small,  then 

V2 

e-  ^  =  -   — 7 (47) 

2  gm2 

Ruehlmann    then    introduces    further    quantities    as    follows  : 
/  =  length  of  canal  a^which  uniform  flow  is  reached ;  hn  =  absolute   :    - 
slope  of  the  surface  KD  and  of  the  canal  bottom  AE\  rj  =  actual   - 
fall  at  section  DE,  below  the  water  level  of  the  inlet]_a  =  sectional 
area,  and  w  the  wetted  perimeter  at  the  section  DE.     Then  the 
required  relative  slope  ratio  for  the  flume,  expressed  as  a  function 
of  the  head  (e  —  e^),  becomes, 

hn  _  r)-(e-e,) 

T          /         (48) 

V*        w  jV2 
and  '-T^  +  a'-F (49) 

wherein  k  is  the  experimental  coefficient  in  Chezy's  well-known 
formula  for  flow  in  rivers  and  canals.  This  formula  is 


Li    a      h 

v  =  k\  -  .  - 

Y  w     I 


(So) 


Ruehlmann  thus  advocates  Eqs.   (47),   (48)  and   (49),  for  the 
solution  of  all  practical  problems  of  the  kind  here  considered. 


54  HYDRAULICS 

Regarding  these  formulae  the  criticism  is  made  that  they  do  not 
take  account  of  section,  slope  and  character  of  the  feeding  river, 
which  factors  determine  the  velocity  of  approach  v,  as  seen  from 
Eq.  (46)  when  solved  for  V.  Thus 


....     (51) 


by  which  V  is  expressed  entirely  as  a  function  of  (e  —  ej  and  the 

velocity  height  —  .     Formula  (46)  is,  therefore,  in  error. 

2g  _  _ 

The  water,  in  flowing  from  the  section  AC  to  the  section  DE,  is 
subjected  to  a  variable  velocity.  Then  for  the  case  of  an  accelera- 
tion,  the  surface  might  be  represented  by  the  dotted  line  CD.  For 
a  retardation  this  surface  would  be  the  dotted  line  CiD,  while  the^ 
special  surface  CiKD  II  AE  could  scarcely  be  expected.  VVXAY 

The  distance  /,  required  for  the  velocity  of  discharge  V  to  become 
constant  and  equal  to  the  velocity  in  the  flume,  also  the  total  fall  y, 
cannot,  in  the  light  of  our  present  knowledge  of  hydraulics,  be 
computed  with  any  degree  of  precision.  These  factors  can  be 
ascertained  only  for  an  existing  flume  by  means  of  carefully  con- 
ducted current  meter  observations  and  slope  determinations. 

w     V2 
The  second  term  -  /  —  ,  of  Eq.  (49),  presupposes  that  the  sur- 

d          KT 

face  KD  i,  AE  and  that  the  mean  velocity  over  the  distance  /  is 
constantly  equal  to  V.  Both  of  these  presumptions  are  certainly 
incorrect  and,  therefore,  Eq.  (49)  is  not  strictly  reliable. 

In  presenting  the  case  of  a  flume  with  sluice  gate  at  the  inlet 
from  the  river,  Ruehlmann  employs  the  experiments  made  by 
Lesbros  on  small  troughs  of  o  .  2  meter  in  width,  placed  in  the  pro- 
longed axis  of  the  supply  channel,  which  latter  was  3  .  68  meters 
wide,  with  a  supply  orifice  o  .  2  meter  wide.  The  troughs  were  3 
meters  long  and  the  experiments  cover  a  range  of  slope  from 
i  :  20  to  i  :  2.9. 

Owing  to  the  large  dimensions  of  the  supply  channel  as 
compared  with  those  of  the  trough,  the  velocity  of  approach  was 


FLOW  IN  RIVERS  AND   CANALS 


55 


necessarily  very  small.     Lesbros   employs  the  formula  for  dis- 
charge through  a  lateral  orifice  into  air  and  finds  for  his  experiments 


(52) 


wherein  H  =  the  fall  between  the  upper  level  and  the  sill  of  the 
orifice  and  h  =  the  same  fall  to  the  upper  edge  of  the  orifice  or 
lower  edge  of  the  gate.  From  this  formula  Lesbros  computes  the 
values  for  p  which  satisfy  all  the  conditions  of  his  experiments. 

While  it  must  be  admitted  that  this  work  might  serve  a  useful 
purpose  within  the  scope  of  the  observations,  yet  it  is  apparent  that 
the  above  formula  (52)  is  not  applicable  to  the  case  nor  can  the 
discharge  coefficients  thus  found  be  utilized  for  computations 
relating  to  large  flumes  or  canals. 

Ruehlmann  (p.  269,  §  104)  presents  another  formula  for  the 
case  illustrated  in  Fig.  25,  where  an  obstruction  is  interposed  at  E, 
thus  slightly  damming  the  water.  The  area  av  with  water  level  rj 
above  the  center  of  the  opening  at  MN,  is  supposed  to  be  located 


Fig.  25 


at  such  a  point  where  the  parallelism  of  the  filaments  of  flow  is 
again  established.  Taking  areas,  velocities  and  heights  as  indi- 
cated in  the  figure,  then  from  the  principle  of  the  conservation  of 
energy,  the  following  equation  may  be  written  out  : 


or 


2g 


•    •     (53) 


56  HYDRAULICS 

Herein  M  represents  the  mass  of  water  passing  any  section  per 
second. 

For  the  case  where  A  is  very  large  in  comparison  with  a,  Poncelet 
first  developed  the  following  formula  from  Eq.  (53) : 


i  + 


(A - 

\aa 


(54) 


In  the  preceding  chapters,  on  the  derivation  of  new  weir  formulae, 
it  is  conclusively  shown  that  the  energy  stored  in  a  moving 
mass  of  water  is  not  expended  entirely  in  the  production  of  flow 
through  a  contracted  orifice,  but  that  a  large  portion  of  that  energy 
is  lost  as  impact  against  all  objects  causing  obstruction  or  contrac- 
tion, and  hence  the  principle  of  the  conservation  oj  energy  is  not 
applicable  to  a  mass  of  water  approaching  a  sluice  or  orifice. 

By  inspection  of  Eq.  (53)  it  is  readily  seen  that  the  difference  in 
height  (h  —  ^)  furnishes  the  pressure  height  producing  the  accel- 
eration necessary  to  increase  v  to  the  value  of  Fx  and  also  to  over- 
come the  resistance  encountered  in  reducing  V  to  the  value  F,. 

/  F2       v2\ 
Also,  the  expression  [  —  - )  is  incorrect  because  the  velocities 

\2g         2gl 

V 'j  and  v  are  in  no  wise  related,  since  v  is  first  accelerated  to  the 
value  V  and  is  then  retarded  to  the  value  Fr  Furthermore,  the 
pressure  height  necessary  for  the  acceleration  of  v  becomes  greater 
in  proportion  to  the  increased  resistance  to  flow  through  the  area  a 
due  to  details  of  design.  And  lastly,  the  velocity  Ft  is  not  depend- 
ent entirely  on  the  pressure  height  (h  —  ??),  but  largely  on  the 
slope,  width,  depth  and  frictional  resistances  in  the  discharging 
flume.  Hence  for  two  different  cases  even  though  v  and  V1 
remained  the  same,  the  controlling  height  (h  -  vj)  might  be  very 
different,  a  condition  which  cannot  be  rectified  in  the  equation  as 
it  stands. 

The  last  expression  I-  ^-M  from  Eq.  (53),  represents  the 
pressure  height  corresponding  to  a  retardation  F  --  V1  for  the 


FLOW  IN  RIVERS  AND   CANALS  57 

case  of  a  sudden  contraction  in  a  closed  pipe,  according  to  Carnot's 
principle.  However,  this  principle  is  not  applicable  to  open 
flumes,  because  the  water  is  not  confined  on  all  sides  and  at  least 
a  portion  of  the  head  (h  —  y)  can  be  expended  in  raising  the  level 
of  the  discharge  surface.  The  velocity  V  is  not  suddenly  changed 
to  V1  but  the  change  is  gradual  and  hence  little  of  the  vis  viva 
is  lost.  All  this  is  contrary  to  the  underlying  principles  of  the 
formula,  and  hence  both  Eqs.  (53)  and  (54),  are  wrong  and 
irrational. 

2.    Proposed  General  Solution. 

It  is  thus  seen  that  no  reliable  formulae  exist  for  any  of  the  cases 
of  flow  just  discussed. 

However,  such  problems  can  be  solved  with  reasonable  accuracy 
by  employing  the  formulae  previously  derived  for  submerged  and 
sluice  weirs  in  combination  with  such  formulae  as  those  of  Ganguillet 
and  Kutter  for  flow  of  water  in  rivers  and  canals. 

Thus  the  problem  presented  in  the  above  Fig.  25,  may  be  solved 
with  the  aid  of  Eqs.  (36),  by  taking  the  highest  point  of  the  dis- 
charge surface  in  the  flume  as  the  governing  level  of  the  lower 
pool  and  assuming  the  velocity  of  flow  for  this  point  equal  to  Vr 

In  the  derivation  of  the  previous  formulae  it  was  seen  that  the 
water  in  the  lower  pool  influences  the  discharge  quantity  by  exerting 
a  hydrostatic  counterpressure,  and  also  by  producing  a  suction 
on  the  area  of  flow.  Hence  the  new  formulae,  which  make  allow- 
ances for  these  various  conditions,  are  in  every  way  applicable  to  a 
flume  supplied  from  a  lake  or  river,  provided  the  depth  and  velocity 
of  the  water  discharged  thror  ^h  the  flume  is  previously  known. 

By  referring  to  Eqs.  (36),  and'  uie  reasoning  there  followed,  it  is 
understood  that  the  total  head  H 2,  between  the  upper  and  lower 
pools  adjacent  to  a  sluice  gate,  is  not  the  governing  factor  deter- 
mining the  discharge  through  the  flume,  but  that  the  quantity  of 
water  discharged  is  determined  by  the  dimensions  and  slope  of  the 
flume  or  canal.  Hence,  the  formulae  for  flow  in  r*  -rs  and  canals 
must  be  employed  for  the  purpose  of  deciding  the  nee  >ssary  dimen- 


$8  HYDRAULICS 

sions  and  slope  to  deliver  the  required  quantity,  and  this  result  is 
then  used  in  the  new  formulae  for  flow  through  regulating  works. 

From  many  gagings  which  have  been  conducted  in  the  past, 
formulae  of  more  or  less  accuracy  are  now  available  for  flow  through 
rivers,  creeks,  canals  and  flumes.  These  express  the  velocity  or 
quantity  of  flow  in  terms  of  cross-section,  slope  and  friction. 

For  steep  slopes  in  canals  or  flumes,  Chezy's  formula  is  probably 
most  applicable.  This  formula,  as  revised  by  Bazin,  in  1897,  is 


=  AV  =  ACVrs,    where  C  = 22 for  feet, 


and  C  = 


(55) 


and  V  =  velocity  in  feet  per  second ;  r  =  mean  hydraulic  radius  in 

^ 
feet  =  —  =  area  of  flow  divided  by  the  wetted  perimeter;  5  =  slope 

w 

=  fall,  in  feet,  divided  by  length,  in  feet,  over  which  the  fall  occurs. 
The  experience  coefficient  m,  depending  on  the  roughness  of  the 
wetted  surface,  is  given  by  Bazin  (1897)  as  follows: 

For  smooth  cement  or  planed  wood    .    .    .    .   m  =  0.06 

For  rough  planks  and  brick m  =  0.16 

For  masonry m  =  o .  46 

For  regular  earth  beds  and  slopes m  =  0.85 

For  canals  in  good  order .-    .    .   m  =  1.30 

For  canals  in  very  bad  order m  =  i .  75 

These  values  are  applicable  to  dimensions  in  feet  and  meters 
alike. 

For  tabulated  values  of  C,  corresponding  to  values  of  r  from 
one  to  ten,  see  tables  46  and  47,  p.  565,  Merriman's  Hydraulics, 
1904. 

Kutter's  formula  for  the  value  of  C  in  Eqs.  (55)  is    probably 


FLOW  IN  RIVERS  AND   CANALS  59 

the  best  in  the  present  state  of  our  knowledge  for  small  slopes. 
This  value,  for  r  and  V  in  feet,  is 


....    (56) 


n   I       ,          < 
i  +  —(41.65  +  - 

Vr\  s 

For  r  and  V  in  meters  the  formula  becomes 


(57) 


s       I 

The  values  of  n  in  both  formulae  (56)  and  (57)  are  alike,  and 
according  to  Kutter  they  are  as  follows : 

For  well  planed  timber n  =  0.009 

For  neat  cement n  =  o.oio 

For  cement  with  one  third  sand n  =  o.on 

For  unplaned  timber n  =  0.012 

For  ashlar  and  brick  work n  =  0.013 

For  unclean  sewers  and  conduits       n  =  0.015 

For  rubble  masonry n  =  0.017 

For  canals  in  very  fine  gravel n  =  0.020 

For  canals  and  rivers  free  from  stones  and  weeds  n  =  o. 025 

For  canals  and  rivers  with  some  stones  and  weeds  n  =  o .  030 

For  canals  and  rivers  in  bad  order n  —  0.035 

Tabulated  values  for  C,  by  Kutter's  formulae,  are  given  in 
Tables  44  and  45,  p.  564,  Merriman's  Hydraulics,  1904;  also  in 
Trantwine,  but  most  completely  in  Bellasis,  p.  183  et  seq. 

With  the  use  of  either  Bazin's  or  Kutter's  formulae  then,  the 
values  of  Q  and  V  can  be  determined  for  any  particular  flume  or 
canal.  Or,  suppose  it  is  desired  to  design  a  flume  to  deliver  an 
assigned  quantity  of  water  Q,  then  the  required  slope,  the  area 


6o  HYDRAULICS 

and  the  velocity  may  be  so  chosen  as  to  fulfil  the  conditions  for 
discharge. 

After  thus  assigning  dimensions  and  slope  to  the  flume,  then 
the  discharge  area  and  head,  necessary  in  the  regulating  works  at 
the  flume  inlet,  can  be  determined  from  one  of  the  previously  given 
weir  or  sluice  gate  formulae.  Should  the  head  H2,  between  the 
supply  canal  and  the  flume,  be  given,  then  the  discharge  area  alone 
will  require  dimensioning. 

If  the  problem  be  stated  thus  :  Given  a  regulating  sluice  gate  of 
certain  dimensions  at  the  inlet  of  a  flume,  the  slope  s  and  width  B 
of  the  latter  being  also  given,  to  find  the  depth  d,  and  quantity  of 
discharge  Q,  through  the  flume,  then  the  following  solution  is  pro- 
posed :  First  compute  the  discharge  through  the  sluice  gate,  using 
one  of  the  formulae  30,  36  or  38  appropriate  to  the  case  in  hand, 
and  assuming  that  the  discharge  takes  place  into  the  air  without 
any  water  in  the  flume.  This  value  is  naturally  excessive  and 
hence  it  can  be  reduced  somewhat  to  obtain  the  first  approximate 

value  Qv  which,  when  inserted  into  the  formulae  (55),  will  give 

^4 
preliminary  values  A1  and  Ft  and  thus  obtain  a  depth  dl  =  —  '• 

Now  by  inserting  the  values  A  t  and  V1  into  the  original  sluice  gate 
or  weir  formula  first  employed,  a  second  quantity  Q2  is  found  as 
a  second  approximation. 

By  a  repetition  of  this  process,  using  Q2  in  the  Chezy  formula 

^4 
to  obtain  A2  and  V2  and  finding  d2=  —?  ,  and  then  employing  A2 

and  V2  in  the  sluice  gate  formula  to  find  a  value  <23,  finally  obtain 
such  a  value  Q  as  will  satisfy  both  equations,  from  which 


* 


Q       A 
= 


is  found  with  reasonable  accuracy. 

Perhaps  a  better  way  to  carry  on  this  successive  approximation 
is  to  plot  the  curves  representing  the  general  relation  between  Q 
and  dj  for  each  of  the  two  formulae,  and  these  two  curves  will 


FLOW  IN  RIVERS  AND   CANALS 


6l 


intersect  in  a  common  point,  the  Q  and  d  of  which  will  satisfy 
both  equations. 

The  foregoing  discussion  has  been  confined  to  cases  where  the 
axis  of  the  flume  is  in  continuation  of  the  supply  channel  axis  and 
the  water  in  the  flume  extends  upstream  to  the  regulating  works. 

3.  New  formula  for  the  case  when  all  the  available  water  of  a 
lake  or  river  should  be  diverted  into  a  lateral  canal  or  flume  by  a 
dam  built  normally  to  the  river. 

A  factory  flume  is  required  to  furnish  an  assigned  quantity  Q, 
to  be  supplied  from  a  river  by  interposing  a  dam.  The  flume  is 
supposed  to  have  its  inlet  just  above  the  weir.  This  will  require 
assigning  dimensions  to  the  regulating  gate  at  the  flume  inlet  and 
also  to  the  flume  itself  and  then  give  the  flume  such  a  slope  as  is 
necessary  to  deliver  the  quantity  Q. 

Formulae  for  the  solution  of  such  problems  have  not  hitherto 
been  proposed,  hence  the  following  discussion  is  offered. 


/    X  \  /'«  n 

-E  

NN'/ 

r 

F  

^ 

G 
D                                                        M 

"x-x^s  -Xxv..;.,:;'  "  "      ,.         N..     1     .,  

h 

\ 

D 

^ZZ2$i$iii& 

Fig.  26 

In  Fig.  26,  let  EF  represent  a  spillway,  with_  regulating  gate  at 
GM ,  both  normal  to  the  flow  of  the  river.  JK  is  a  lateral  diver- 
sion channel  making  the  angle  6  with  the  axis  of  the  river. 


62 


HYDRAULICS 


A  longitudinal  section  along  A  A,  showing  the  flume  inlet  jja^a, 
is  given  in  Fig.  27.  The  first  case  to  receive  consideration  is  when 
the  water  level  of  the  river  is  even  with  the  crest  of  the  dam  at  E, 
and  the  entire  discharge  Q  passes  out  through  the  flume. 


Fig    27 

The  hydrostatic  pressure  against  the  flume  area  is  taken  the 
same  as  against  the  dam,   and  the  hydrodynamic  pressure  as 

found  from  Eq.  (13)  =  p5  =  7  —  Bk  which   (per  unit  of  area) 

o 


V  -. 
g 


becomes  p6 

Using  the  dimensions  indicated  in  the  twojfigures,  then  ai  =  ae 
sin  0,  and  the  dynamic  pressure  on  the  area  ai  is  all  that  is  trans- 
mitted to  the  discharge  area.  Hence  the  pressure  on  this  entire 

area  ^ 

=  p=y-  bld  sin  6. 
g 

Let  the  line  Jr  represent^  the  magnitude  of  p7,  which  may  be 
resolved  into  components  Jg  and  Ja.  Then  the  component  Jg, 
normal  to  ae,  is  found  from: 

tf 
g 


Jg  =  Jr  cos  (90  —  6} 


which  may  be  represented  by  a  rectangular  prism  of  height  d,  in 
front  of  the  discharge  area.     This  prism  will  have  a  breadth  b' ', 

and  length  —   sin2  6. 
g 


FLOW  IN  RIVERS  AND   CANALS  63 

The  hydrodynamic  pressure  pe  of  the  approaching  water,  acting 
normally  to  the  banks,  and  hence  deflected  into  the  direction  of 
the  flume,  maybe  regarded  active  over  the  area  b'd  and  in  the 
direction  JK. 

The  hydrodynamic  pressure  deflected  by  the  weir  against  the 
flume  inlet,  by  analogy  to  Eq.  (n),  becomes 


p 


=  0.25 


because  H  is  now  d,  —  =  -  becomes  0.5  &',  and  <j>  =  90°.     This 

pressure  p4,  being  equally  distributed  over  the  area  b'd,  may  again 
be  represented  by  a  rectangular  prism  in  front  of  the  discharge 

v* 
area.    Assume  its  height  d,  breadth  V  and  length  0.25  -• 

o 

The  water  flowing  in  the  lower  section  of  the  river,  exerts  a 
dynamic  pressure  which  is  transmitted  normally  and  horizontally 
against  the  flow  area,  and  may  be  represented  by  a  triangular 
prism  of  height  (k  —  d),  a  width  =  V  sin  0,  and  a  bottom  length  /?, 


' 

:  —  *  —       ~~~7 

| 

^~^^^_ 

4  v  p/—  1 

oil—  o 

i 

"^                                   -f         _^v 

1 
1 

s    *    &  \ 

'      "^                             1 

r-i  rz  r      e 

3  «1   «1 

0  ^^/^li^^ff^^^^^^/fi^///^//^^/^ 

i 

MUWJ!^ 

1 

Fig.  2S 

which  may  be  determined  from  Eq.  (16)  as  follows:  see  Figs.  27 
and  28. 


From  Eq.   (16)  p  = 


_ 
represented  by  oa^  =  T.  --  ,  in  Fig.  28,  and  -  is  equal   to 


Tr  cos2  -^  ,  in  which  the  area  VH  is 
o'gti          2 


64  HYDRAULICS 

om.  Also  the  area  Bk  now_becomes  (k  —  d)  sin  0,  and  since  the 
resultant  in  the  direction  JK  is  wanted,  the  whole  expression,  for 
a  unit  width,  finally  becomes 

'*-  •  ••  <s8> 


All  of  these  prisms  are  now  combined  in  Fig.  28,  as  follows: 

_        ^  _  ^2     _         ny2  _ 

make  ae  =  -sin2  0;  ee.  =  o.  25  -  ;  0w  =  --  ,  and  8  =  rr_;  then 

£  £  2  J? 

o  o  o 

the  area  e2es#r  represents  the  hydrostatic  pressure  and  the  final 
hydraulic  pressure,  active  on  the  area  a^e^e,  is  represented  by  the 
area  ae2pr2alf 

The  resultant  unit  pressures  along  any  horizontal  filament  of  the 
pressure  area  are  thus  easily  found  from  Fig.  28,  and  the  following 
values  are  now  derived: 

The  hydraulic  pressure  along  the  surface  filament  is 

y  S  =  ae  +  ee2  =  —  (0.25  +  sin2  6)    .     .     .     (590) 

o 

The  hydraulic  pressure  along  op,  where  the  hydrostatic  counter- 
pressure  of  depth  7\  becomes  active,  is 


S,  =  o^+^~p  =  S  +  d-^  =  S  +  d-(T1-^~  \    (596) 
The  hydraulic  pressure  at  the  bottom  of  the  flume  is 


Employing  these  values  of  S,  Sv  and  52,  the  discharge  into  free 
air,  through  the  upper  portion  ao  of  the  section,  may  now  be  found 


from  Eq.  (19),  in  which  H  =  d  —  T.  +  -  , 

2  g 

and  si-S  =  d-T1  +  —  , 

2  g 


hence  Ql  =  f 


FLOW  IN  RIVERS  AND   CANALS  65 

In  like  manner  the  discharge  through  the  submerged  portion 


oa  =  T.  —  —  —  ,  is   found   from  Eq.  (19)  by  making  H  =  oav 

2  g 
then 


The  total  flow  through  the  section  Ifd  =  a^ae^e  is  then 

e  =  QI  +  e, 


Since  the  Chezy  formula  also  furnishes  a  value  for  the  uniform 
flow  through  the  flume,  then  for  any  given  case,  for  which  the 
dimensions  are  assigned  or  determined  from  local  conditions,  the 
three  unknowns  can  be  found  as  previously  indicated. 

When  the  flume  is  supplied  from  a  lake,  or  river  of  sluggish  flow, 
then  the  velocity  of  approach  can  be  disregarded  and  the  above 
formulae  (59)  reduce  to  simpler  forms.  Thus  by  making  v  =  o, 
then 


n 
5  =  o,    S2  =  Sl  =  d  —  Tl  H  --  ,  and  approximating  Q2  as  was 

done  in  Eq.  (28/),  then 


Formulae  (59)  will  also  apply  to  the  case  illustrated  'in  Fig.  9, 
because  when  no  discharge  takes  place  over  the  spillway,  the 
approaching  flow  exerts  equal  pressure  in  all  directions. 

The  application  of  Eqs.  (59)  will  now  be  illustrated  by  solving  a 
problem  such  as  is  likely  to  occur  in  practice.  The  dimensions  are 
all  taken  in  the  metric  system,  merely  as  a  matter  of  convenience. 


66  HYDRAULICS 

Problem.  Given  a  river  10  m.  wide,  discharging  6  m.3  per 
second.  It  is  desired  to  build  a  factory  flume  having  its  inlet 
just  above  a  spillway  which  is  to  be  built  normally  to  the  river 
and  of  3  m.  height  with  vertical  upstream  face.  The  flume  is  to 
make  an  angle  of  45  degrees  with  the  axis  of  the  river  and  must 
deliver  the  full  river  discharge  to  a  factory,  2000  m.  distant,  when 
the  water  stands  level  with  the  top  of  the  spillway. 

The  section  of  the  flume  may  be  chosen  as  trapezoidal  with  i  :  i 
slopes  and  of  such  area  that  when  the  water  is  flowing  with  velocity 
V  =  i  m.  per  second,  it  shall  deliver  the  required  Q  when  the 
depth  of  flow  7\,  in  the  flume,  is  i  m. 

The  question  then  resolves  itself  into  finding  the  width  &',  and 
depth  d  of  the  flume  inlet,  also  the  bottom  width  &",  and  slope  5  of 
the  flume,  necessary  to  discharge  6  m.3  per  second. 

The  terms  in  Eqs.   (59)  will  then  have  the  following  values: 

B  =  10  m.,  Q  =  6  m.3,  k  =  3  m.,  v  =  -^  =0.2  m.  per  second, 

Bk 

V  =  i  m.  per  second,  2\  =  i  m.,  and  6  =  45°.  Also  ^r  =  90° 
when  the  river  bank  at  the  flume  inlet  is  made  vertical  by  a 
retaining  wall. 

The  four  unknown  quantities  are  b'  ',  d,  b"  and  s,  but  as  only 
three  equations  are  available  it  will  be  necessary  to  make  b'  =  b" 
which  is  equivalent  to  making  the  bottom  width  of  the  flume  equal 
to  the  uniform  width  of  the  flume  inlet. 

From  Q=T1(b'+  7\)  V  the  value  b'=  -2--  T,=  5  m.  =  6." 
The  slope  s,  of  the  flume,  is  found  by  Chezy's  formula,  Eq.  (55), 


wherein  r=       2    o  n       =  0.7665  m.  and  Vr  =  0.875.     Taking 
7.828 


6  x        x 


=  0.0326  or  s  =  0.001063,  and  the  total  fall  required  in  the 
flume  over  the  distance  of  2000  m.  will  be  2.  12  m.     By  building 


FLOW  IN  RIVERS  AND   CANALS  67 

the  flume  rectangular  in  section,  with  cement  walls  and  bottom, 
to  reduce  friction,  and  maintaining  the  area  6  m.3  =  i  X  6,  and 
velocity  V  =  i  m.  per  second,  this  total  fall  could  be  reduced  to 
about  0.4  m. 

Lastly,  to  find  d  from  the  formulae  (59)  the  coefficients  JJL  and  /^ 
must  be  known.  In  want  of  better  data,  these  values  are  assumed 
each  equal  to  0.6,  a  value  which  is  based  on  the  experiments  of 
Francis. 

Now  substituting  all  of  the  foregoing  values  into  the  Eqs.  (59) 
it  will  be  found  that  the  value  d  =  i .  136  m.  satisfies  the  three 
equations,  making  Q1  =  0.64  m.3,  Q2  =  5.36  m.3,  and  Q  =  Ql 
+  <22  =  6  m.3 

Hence,  if  the  flume  inlet  be  made  5  m.  wide  X  1.136  m.  deep, 
thus  making  the  water  in  the  flume  d  —Tl  =  0.136  m.  lower 
than  in  the  river,  then  the  flume  will  discharge  the  required  quantity 
Q  =  6  m3. 

This  example  illustrates  the  points  previously  brought  out  in 
the  present  chapter,  viz.,  that  the  slope  of  the  flume  is  not  a 
function  of  d  —  7\,  but  depends  on  the  section  and  roughness  of 
the  flume. 

4.  New  formula  jor  the  case  when  a  portion  of  the  flow  in  a  river 
should  be  diverted  and  the  remainder  be  discharged  over  a  weir  built 
normally  to  the  river. 

This  case  is  exactly  like  the  preceding,  only  that  a  portion  of  the 
water  is  discharged  over  the  weir  along  a  profile  A^E^  Fig.  27, 
such  that  the  depth  H  over  the  crest  of  the  weir,  may  be  main- 
tained aside  from  furnishing  a  stipulated  flow  through  the  power 
canal.  Q  will  here  represent  the  quantity  to  be  diverted  for  power 
purposes. 

The  problem  of  finding  the  hydraulic  pressure  active  on  the 
flume  area  alaJJ2  is  very  difficult,  owing  to  the  interference  of 
cross  currents,  the  retarding  effect  of  which  would  depend  largely 
on  the  relative  quantities  of  flow  in  the  two  directions.  Hence 
nothing  better  than  an  approximation  could  be  expected. 


68 


HYDRAULICS 


The  hydrostatic  and  hydrodynamic  pressures  exerted  against 
the  area  of  flow  into  the  flume  will  be  found  as  in  the  previous 
solution  but  modified  as  follows: 

1.  The  flow  of  approach  will  not  expend  its  entire  hydro- 
dynamic  pressure  —  on  the  area  of  flow  because  the  direction  of 
the  river  current  is  past  this  area. 

2.  For  this  same  reason,  no  hydrodynamic  pressure  will  be 
deflected  from  the  river  banks  against  the  area  of  flow. 

3.  That  portion  of  the  hydrodynamic  pressure  resulting  from 
the  flow  against  the  area  a^JJ^  Fig.  27,  will  now  be  divided  so 
that  in  the  present  case  this  pressure  will  be  assumed  only  half  as 
great  as  it  was  with  the  water  dammed. 


Fig.  29 

These  several  conditions  are  shown  on  the  profile,  Fig.  29,  where 
the  graphic  areas  represent  the  pressures,  exerted  by  various  causes, 
against  the  area  of  flow.  Figs.  26  and  27  still  apply. 

The  depth  of  flow  into  the  flume  inlet  is  now  d  +  H,  Fig.  27; 
the  suction  on  the  flow  area  produced  by  the  water  in  the  flume  is 

represented  by  om  =   -  -  ,  and  the  uniform  depth  of  flow  in  the 
flume  is  7\.     The  pressure  height  a2o  is 


ano 


FLOW  IN  RIVERS  AND   CANALS  69 

from  which  the  quantity  Qlt  flowing  through  the  discharge  area 
a2o  into  free  air,  is  found  as 


In  the  derivation  of  Eq.  (15),  the  hydraulic  pressure  exerted  by 
the  water  flowing  below  the  discharge  section  against  this  section, 

tf  j, 

was  found  to  be  pQ  =  7  —  Bk  cos2  —  .     For  the  present  case  k  —  d 

g  2 

must  be  substituted  for  k,  and  b  sin  0  for  B.    Also  the  hydro- 

v2 
dynamic  pressure  becomes  —  and  retaining  ^  as  the  angle  of  slope 

o 

of  the  river  bank  below  the  flume  inlet,  then  the  value  pe  becomes 
p  '  =  7  -^  (k  -  d)  V  sin  6  cos2  *  . 

2  g  2 

Since  this  pressure  may  be  assumed  as  varying  with  the  depth, 
being  a  maximum  at  the  bottom  of  the  flume  alev  and  zero  at  pm, 
it  may  be  represented  by  a  triangular  prism  of  base  /?,  height 

-  f  Tl  ---  J  and  length  V.     The  previous  equation  for  pj  gives 


Hence  the  trapezoid  opalr1  represents  the  total  effective  hydraulic 
pressure  against  the  area  of  discharge  ~oa^  into  the  flume.  The 
pressure  Sl  along  the  filament  op  and  the  pressure  S2  along  the 
bottom  of  the  flume  alrl  are  evaluated  as 

<wF2 

5,  =  d  +  H  +  -         -  7\     .     . (620) 


k-d 


Jsinflcos2^  .     .    (626) 


70  HYDRAULICS 

The  quantity  of  discharge  Q2  through  the  area  oav  Fig.  29, 
is  found  from  Eq.  (19)  by  making  the  head  equal  to  ( Tl—  -  -V 
thus: 


(620 


Hence  Q   =  Ql  +  Q2    ...........    (62^) 

Should  v  and  k  —  d  be  small,  then  the  value  for  Q2  might  be 
reduced  to  the  form 


These  formulas  (62),  in  combination  with  the  Chezy  formula, 
furnish  a  solution  for  problems  of  the  kind  previously  treated  when 
only  three  unknown  quantities  remain  to  be  determined. 

5.  New  'formula  for  the  case  when  a  portion  of  the  flow  in  a  river 
should  be  diverted,  and  the  remainder  be  discharged  over  a  weir  built 
diagonally  across  the  river. 

This  case  is  illustrated  in  plan,  Fig.  9,  while  Fig.  27  will  again 
serve  for  longitudinal  section.  Finally  Fig.  30  shows  the  pressure 
areas  active  against  the  discharge  area  of  the  flume.  _ 

The  water  in  the  river  approaches  with  a  surface  A^E2  and  the 
depth  on  the  crest  of  the  weir  is  again  taken  equal  to  H,  see  Fig.  27. 

From  Fig.  9,  the  totaj  hydrodynamic  pressure  of  the  flow,  over 

the  height  k  of  the  weir,  is  <yBk  —  ,  represented  in  the  figure  by 

o 

the  magnitude  e^av  which  is  resolved  into  components  e1g1  and 
7jv  respectively  normal  and  parallel  to  the  face  of  the  weir.  Then 
7^gl  is  expended  against  the  weir  and  eJ1  becomes  effective  on  the 
discharge  through  the  flume  in  an  amount 

v2 


a^  cos  <j>  =  ejl  =  7  -  Bk  cos  </>, 

o 


directed  towards  the  bank  LM. 


FLOW  IN  RIVERS  AND   CANALS  ^\ 

That  portion  p  of  the  pressure  ejlt  Fig.  9,  which  is  active  over 
the  height  d,  Fig.  27,  of  the  flume  inlet,  is  found  from  Fig.  26  as 


„, 

p  =  y  -   dbf  - 

g         sm 


by    observing  that  LM  = and  MN  =  - — -  from   Fig.   o. 

tan  0  sin  0 

But  the  pressure  exerted  normally  to  b',  Fig.  26,  is 

p  cos  (0  —  0)  =  7  —  db'  — — -n  cos  (0  —  0). 
g         sin  0 

Now  since  p  is  uniformly  distributed  in  the  vertical  direction, 
it  may  be  represented  by  a  rectangular  pressure  area  of 

length  &',  height  d,  and  breadth  /?  =  -  •  ^-^  cos  (0  -  0)     (63) 

g      sin  0 

acting  against  the  flume  inlet. 

The  hydrodynamic  pressure  exerted  against  the  weir  surface 
below  the  flume  inlet  is  expressed  by 

v* 
1          g 

and  the  portion  of  p^  expended  over  the  width  of  the  inlet  03/3,  Fig. 
27,  against  the  river  bank  is 

v2  ,,  ,  sin 


This  pressure  p2  acting  against  an  inclined  bank,  making  the 
angle  ^  with  the  horizontal,  causes  a  pressure  against  the  flume 
discharge  area  tending  to  accelerate  the  discharge  velocity.  The 
amount  of  this  effect  is  given  from  Eq.  (15)  as 


/L      j\  T./  sn          2  ir 
p  =  y  -  (k  —  d)b'  T—^  cos'  —  • 
g  sin  d          2 

Since  />6  is  not  uniformly  active  but  creates  maximum  effects  at 
the  bottom  of  the  flume,  it  is  again  represented  by  a  triangular 


HYDRAULICS 


pressure  area  against  the  discharge  opening.     This  pressure  prism 
will  have  a  length  &',  height  (  J'1 j  and  base  /?x  found  from 


b'  /  ' 

-  (T  l  - 
2  \ 


2g 

7y  mn 

7  —  (k  —  d)  b'   - — '-  cos"  —  > 
g  sm  6  2 


whence 


The  pressure  areas,  effective  against  the  flume  inlet,  are  graphic 
ally  indicated  in  Fig.  30. 


Ai 


Fig.  30 

The  suction  produced  by  the  discharge  velocity  V  on  the  section 


_  _      n  _  _ 

a^2  is  represented  by  om  =  -  •  .    The  area  aiaj  represents  the 

hydrodynamic  pressure  active  on  the  flume  discharge  area,  where 
ai  =  aj  =  /?  from  Eq.  (63).  The  trapezoid  #3/r^  represents  the 

nV2 
total  hydrostatic  pressure  plus  the  suction  -  active  on  the  dis- 

charge area.  Also,  making  rrl  =  /?15  the  triangle  prr^  becomes 
the  pressure  area  resulting  from  the  flow  below  the  bottom  of  the 
flume.  Finally,  transferring  the  triangle  a3ei,  representing  the 
hydrostatic  pressure  due  to  the  head  H,  to  the  position  a^e^a  in 
front  of  the  discharge  area,  the  whole  hydraulic  pressure  will  be 
represented  by  the  figure  a^ 


FLOW  IN  RIVERS  AND   CANALS  73 

Since  the  pressure  on  the  part  section  a2o,  Fig.  30,  undergoes 
an  abrupt  change  e^e,  which  could  not  exist  in  reality  and  would 
necessitate  some  further  complications  in  the  resulting  formulae, 
therefore,  the  pressure  area  opee^  may  be  replaced  by  the  triangle 
opar  In  this  triangle  the  height  and  base  have  the  following 
values  : 


_       r  ny*         I  _  _      _ 

a2o  =  \d  +  H  +  -  —  Tl     and    op  =  S  =  oil  +  ij)y 

2  g  J 

making 


-7-,     .     (65a) 

The  quantity  of  discharge  Ql  through  the  part  section  a2o  will 

then  be 

2          - 

S.    .     (656) 


Also,   the   resulting   pressure   a/t  =  St  =  a/  +  rrl  =  S  +  plt 
active  at  the  bottom  of  the  flume,  will  be 


c 

-s+  ^ 

k-d   - 

fef)C° 

G2       T                        fAr/'^ 

Pi 

g 

nV2 

S                   .       .        (0$C) 

2g 

The  quantity  of  flow  Q2  through  the  lower  portion  a^o  of  the 
flume  inlet  will  then  become,  after  inserting  the  proper  value  for 
H  into  Eq.  (19), 

•nV2! 


T 

1  i  " 


^-S1)  .    .     (65d) 
3 


The  total  discharge  into  the  flume  is  then 


Should  the  angles  (f>  and  6  be  equal,  or  nearly  so,  and  the  angle  ^ 
be  made  90  degrees,  also  making  v  =  o,  then  these  formulae  (65) 
can  be  very  much  simplified  as  was  done  for  Eqs.  (60). 


74  HYDRAULICS 

By  combining  these  formulae  with  Chezy's  formula,  three 
unknown  quantities  may  again  be  found  as  illustrated  in  previous 
examples. 

If  a  sluice  gate  were  placed  at  the  inlet  to  the  flume  then  the 
hydrostatic  and  hydrodynamic  pressures  resulting  from  the  back 
water  in  the  river  against  the  gate,  may  be  found  in  the  manner 
previously  described,  finally  computing  the  discharge  through  the 
gate  from  one  of  the  formulae  (36)  or  (38)  according  to  the  case 
in  hand. 

This  closes  the  theoretical  portion  of  the  present  treatise. 


CHAPTER  VII. 
EMPIRIC    COEFFICIENTS, 
i.    INTRODUCTORY. 

IN  the  foregoing  theoretical  chapters  no  consideration  was  given 
to  the  various  empiric  coefficients  employed.  This  subject  is  one 
of  vital  importance  to  the  usefulness  of  any  formulae,  which  latter 
can  never  be  •  made  to  include  all  the  disturbing  elements  always 
present  when  dealing  with  river  hydraulics.  The  best  that  can 
ever  be  hoped  for  is  the  adaptation  of  rational  formulae  to  the 
observed  facts  and  to  correct  the  shortcomings  by  the  introduction 
of  numerical  coefficients. 

The  effort  here  made  in  the  direction  of  furnishing  the  rational 
forms  for  many  of  the  complex  problems  frequently  encountered 
in  practice,  by  taking  into  account  nearly  all  of  the  variable  factors, 
should  merit  the  appreciation  of  every  hydraulic  engineer. 

The  attempt  will  now  be  made  to  evaluate  the  coefficients  for 
these  new  formulae  by  employing  all  available  experiments  bearing 
directly  on  the  subject.  However,  suitable  and  reliable  experi- 
ments of  the  kind  in  question  are  not  numerous.  The  very  valua- 
ble and  painstaking  hydraulic  experiments  made  by  Messrs.  A. 
Fteley  and  Frederic  P.  Stearns  and  published  in  1883,  Trans.  Am. 
Soc.  C.E.,Vol.  12,  and  the  classic  "Lowell  Hydraulic  Experiments," 
1855,  by  Mr.  James  B.  Francis,  together  with  some  more  recent, 
but  limited  experiments,  constitute  about  all  the  available  data. 
Thanks  then  to  the  existence  of  these  experiments,  some  reliable 
values  for  p  in  the  new  formulae  have  been  found,  although  in  many 
of  the  cases  theoretically  discussed,  no  empiric  data  are  as  yet  at 
hand,  from  which  to  deduce  coefficients. 

The  rational  formulae,  previously  derived,  will  still  give  values 

75 


OF   THE 

UNIVERSITY 


76  HYDRAULICS 

in  excess  of  the  actual  when  the  coefficients  are  neglected.  Hence 
these  coefficients  are  really  reduction  factors  to  reduce  theoretical 
to  real  discharge.  The  causes  producing  this  effect  may  be  thus 
summarized : 

a.  The  loss  in  its  vis  viva  of  the  flow  of  approach  due  to  impact 
and  deflections  near  and  at  the  discharge  area. 

b.  The  friction  and  cohesion  around  the  wetted  surface  at  the 
discharge  section,  producing  a  retardation  in  the  discharge  velocity. 

c.  The  contraction  in,  and  immediately  adjacent  to,  the  dis- 
charge   section.     This   contraction,  which  is  probably  the  most 
significant  of  the  three  named  effects,  is  variable  and  depends  on 
the  dimensions    of    the   overfall,   on  the   interrelation   of   these 
dimensions,  on  the  canal  and  its  configurations  near  the  weir, 
and  on  the  velocity  and  depth  of  the  flow  of  approach.     In  the 
following,  two  forms  of  contraction  are   distinguished,  complete 
and  partial.     Complete  contraction  takes  place  when  the  discharge 
section  is  suddenly  reduced  on  all  sides  of  the  orifice,  as  for  dis- 
charge from  a  lateral  orifice.     Partial  contraction  occurs  when 
the    approaching    flow  is  confined  only  on  three  sides,  and  the 
discharge  section  is  contracted  on  three  sides. 

The  total  correction  for  all  the  foregoing  retarding  influences 
will  then  be  made  by  introducing  an  empiric  reduction  factor. 
This  factor  for  discharge  into  free  air  will  be  called  //.  and  for 
discharge  through  an  entirely  submerged  section  it  will  be  called  /*r 
For  any  given  set  of  experiments  the  values  of  either  or  both  of 
these  coefficients  are  then  determined  by  inserting  all  the  observed 
quantities  into  the  appropriate  formulae  to  find  the  theoretical 
quantity  of  discharge,  which  latter  divided  into  the  observed  dis- 
charge gives  the  corresponding  ^.  This  method  will  be  applied 
in  the  following  to  all  available  observations  covering  the  widest 
possible  range  in  weir  and  canal  dimensions. 

For  the  most  part  the  observations  from  the  experiments  of 
Messrs.  Francis,  Fteley,  and  Stearns,  have  been  used.  Aside 
from  these  the  Cornell  University  experiments  on  standard  sharp 
crested  weirs  made  in  1899,  and  a  few  old  experiments  made  by 


EMPIRIC  COEFFICIENTS  77 

Lesbros,  in  1829  to  1834,  and  some  made  by  Boileau,  in  1845,  were 
employed  merely  to  extend  the  range  of  the  former  observations. 

It  should  be  noted  that  the  values  ft  are  alike  for  United  States 
and  metric  units,  so  long  as  all  dimensions  are  expressed  in  terms 
of  one  such  unit  only.  That  is,  ft  is  merely  a  ratio  between  two 
volumes,  which  ratio  remains  the  same  independent  of  the  unit 
chosen. 

Regarding  the  coefficient  in  the  formula  for  discharge  through 
lateral  orifices  it  should  be  mentioned  that  the  usually  accepted 
value  is  really  the  product  of  the  numerical  coefficient  f  and  ft,  so 
that  the  theoretical  part  of  the  formula  is 


•Q  =  %bH  V2  gH. 

However,  the  old  values  have  been  so  widely  adopted  that  a 
change  would  greatly  confuse  matters ;  therefore,  in  all  the  follow- 
ing tables  ft  and  f  ft  have  been  tabulated  together. 

2.    COMPLETE    OVERFALLS. 

(a.)  Coefficients  fis  for  Eqs.  (19)  to  (22),  for  weirs  normal  to  the 
channel  and  no  wing  walls,  hence  B  =  b  .  See  observations,  Tables 
II  and  III. 

In  all  these  observations  B  =  b,  $  =  90  degrees,  and  ^  =  90 
degrees  except  for  observations  46  to  50,  where  ^  =  20  degrees  and 
a  wide  crest  was  used.  The  values  for  f  ft  and  ft  are  found  from 
Eqs.  (19)  to  (22)  and  inserted  in  these  Tables  II  and  III. 

Since  it  is  necessary  to  provide  values  for  ft  for  any  particular 
problem  in  hand,  the  following  empiric  formula  (66)  is  offered  for 
its  computation.  In  this  formula,  ft  is  made  to  depend  on  b,  k  and 
H,  and  a  constant  as  expressed  by  Eq.  (66), 

=  -  W  •     (66) 


The  factor  p  is  the  experience  factor  for  any  crest  compared  with 
the  standard  sharp  crest,  and  the  quantity  inside  the  parenthesis 
will  represent  f  fi5  which  is  the  special  value  of  ft  for  sharp  crested 


78  HYDRAULICS 

weirs.  Hence  for  standard  sharp  crested  weirs  p  =  i,  while  for 
all  other  crests,  p  becomes  a  multiplier,  the  value  of  which  depends 
entirely  on  the  nature  of  the  crest. 

The  constants  of  Eq.  (66)  were  derived  from  the  values  f  p 
found  from  the  tabulated  experiments  i  to  22,  Table  II.  When 
inserted  in  Eq.  (66)  and  calling  p  =  i,  because  all  these  experi- 
ments were  made  on  sharp  crested  weirs,  then  for  dimensions  in 
feet: 

+  0.0001326   (67) 


/    H    \  ,  0.00106  7    ,<0. 

0.40105-0.00453  f  J  + — —  -  +  0.00043  b    (68) 


and  for  dimensions  in  meters: 
2 

*' 

The  values  of  f  /*  were  then  computed  from  Eq.  (67)  for  each 
of  the  tabulations  i  to  32.  The  resulting  errors,  expressed  in 
percentage,  were  entered  in  the  last  column  of  Table  II. 

The  tabulations  41-45,  Table  III,  were  similarly  treated  and 
comparatively  close  agreements  were  found  even  though  the  experi- 
ments of  Boileau  were  not  conducted  with  the  same  degree  of 
accuracy,  nor  under  the  same  conditions  as  those  in  Table  II. 

The  tabulations  33  to  40,  in  Table  III,  do  not  fit  Eq.  (67), 
because  the  values  of  k  and  H,  and  in  fact  all  of  the  dimensions, 
are  too  small  to  promise  any  results  which  are  comparable  with 
those  of  Table  II.  However,  the  values  of  f  i*.s  were  found  from 
the  new  formula  (22),*  and  for  these  values  Eq.  (66)  becomes  for 
metric  units: 

(TT      \ 
— — - )  —  0.8738  H  +  0.00048  b   (690) 
±d+k/ 


when  the  observations  33  to  37  only  are  included.     For  the  remain- 
ing three  observations  38  to  40  the  following  equation  was  found : 

(TT          \ 
^j—k)  -0.63  H  +  0.00048  b  .   (696) 

*  The  flume  in  these  experiments  widened  out  suddenly  just  beyond  the  dis- 
charge section,  hence  (b  -f  0.042  H)  was  used  in  place  of  b  in  Eq.  (22)  as  was 
originally  suggested  by  Francis. 


EMPIRIC  COEFFICIENTS 


79 


TABLE  II. 
DETERMINATION  OF  p  FOR  EQS.  (19)  TO  (22)  WHEN  B  =  b.  ANDp=i. 


No. 

Original 
Experi- 
ment 
No. 

Measured  Values. 

Computed 
from  Eqs.  (22) 

Error 
by 
Eq. 
(67) 
Per 
cent. 

B 

b 

k 

H 

V 

Q 

i* 

f*s 

ft. 

ft. 

ft. 

ft. 

ft. 

cu.  ft. 

FROM  TABLE  XHI,  EXPERIMENTS  BY  J.  B.  FRANCIS,  1852. 


I 

67-71 

9.992 

9-995 

5.048 

0.79518 

0.4071 

23.790510.4068 

0.6102 

—  O.  2 

2 

44-5° 

9.992 

9-995 

5.048 

0.97900 

0-5403 

32.56160.4052 

0.6078 

O.O 

3 

51-55 

9.992 

9-995 

5.048 

1.00026 

0.5538 

33.494610.4051 

0.6077 

o.o 

FROM  TABLE  XXVIH,  EXPERIMENTS  BY  FTELEY  AND  STEARNS,  1878. 


4 

i  &  5 

5>° 

5  .  0048 

3-56 

0.1509 

0.054 

1.007 

0.4246 

0.6369 

•  o.o 

5 

6  '  10 

5-o 

5  •  °°44 

3-56 

0.23035 

0.098 

1.8685 

0.4164 

0.6246 

0.0 

6 

ii  '  17 

5-o 

5-oo45 

3-56 

0.33685 

0.168 

3.284 

0.4121 

0.6181 

—  O.I 

7 

18  '  21 

5-o 

5-oo43 

3.56 

0.42425 

0.233 

4-6365 

0.4097 

0.6145 

—  O.I 

8 

22   '  27 

5-o 

5  •  °°49 

3-56 

0.4305 

0.237 

4.736 

0.4096 

0.6143 

—  O.I 

9 

28  '  34 

5-o 

5.0047 

3-56 

0.5116 

0.301 

6.134 

0.4083 

0.6124 

+  0.1 

10 

36 

5-o 

5  .  0046 

3-56 

o-5477 

o.33i 

6.796 

0.4076 

0.6114 

o.o 

ii 

37,41,44 

5-° 

5  •  0040 

3-56 

0.60076 

o-375 

7.8093 

o.  4070 

0.6105 

o.o 

12 

46  &47 

5-o 

5  .  0042 

3-56 

0.69245 

o-455 

9-677 

0.4063 

0.6095 

—  0.  I 

13 

53 

5-o 

5-oo38 

3-56 

0.8047 

0-556 

12.147 

0.4052 

0.6078 

o.o 

FROM  TABLE  XV,  EXPERIMENTS  BY  FTELEY  AND  STEARNS,  1879. 


14 

10 

19.0 

18.997 

6-55 

0.4685 

o.  151 

20.178 

0.4081 

0.6122 

+  0.6 

15 

9 

I9.O 

18.997 

6-55 

0.6460 

0.239 

32  .  685 

o  .  4066 

0.6099 

+  0.4 

16 

8 

I9.O 

18.997 

6-55 

0.8191 

o-334 

46.  760 

0.4058 

0.6087 

+  0.3 

17 

7 

I9.O 

18.997 

6-55 

0-9853 

o-433 

62.023 

0.4065 

0.6097 

o.o 

18 

6 

I9.O 

18.997 

6-55 

0.9873 

o-433 

62  .  061 

0-4055 

0.6083 

—  0.2 

19 

5 

I9.O 

18.997 

6-55 

i.  1456 

o-532 

77.783 

0.4052 

0.6079 

+  0.2 

20 

3 

19.0 

18.997 

6-55 

1.2981 

0.632 

94.192 

0.4055 

0.6082 

0.0 

21 

2 

I9.O 

18.997 

6-55 

1.4546 

o-737 

112.066 

0.4054 

0.6081 

O.O 

22 

I 

19.0 

18.997 

6-55 

1.6038 

0.840 

130.117 

0-4053 

0.6080 

O.O 

FROM  TABLE  XIV,  EXPERIMENTS  BY  FTELEY  AND  STEARNS,  1877. 


23 

30 

5-o 

4.996 

3-i7 

0.0746 

0.023 

0.3652 

0.4450 

0.6675 

+  1.0 

24 

29 

5-o 

4-996 

3-i7 

0.0991 

0.034 

0.5498 

0.4370 

0-6555 

+  0.1 

25 

24 

5-o 

4.998 

3-i7 

o.  1225 

0.046 

0.7526 

0-4345 

0.6518 

-0.9 

26 

20,  21 

5-o 

4-9965 

3-i7 

0.16385 

0.069 

I-I536 

0.4287 

0.6431 

-i-3 

27 

I7»  19 

5-° 

5-o 

3-i7 

0.21826 

0.1033 

I-75073 

0.4219 

0.6328 

—  i.o 

28 

14,  15 

5-o 

4-999 

3-17 

0.25325 

o.  1265 

2-15975 

0.4168 

o.  6252 

—0.4 

29 

12,  13 

5-o 

4.996 

3-17 

0.32605 

o.  180 

3-14475 

0.4143 

0.6215 

-0.6 

3° 

6,  8,  9 

5-o 

4.998 

3-17 

0.48443 

0.3117 

5-696 

0.4112 

0.6168 

-0.7 

3i 

4,5 

5-o 

4.999 

3-i7 

0.6737 

0.488 

9-376 

0.4092 

0.6138 

-0.7 

32 

3 

5-o 

5-o 

3-i7 

0.8118 

0.627 

12.466 

0.4088 

0.6132 

-0.9 

NOTE.  —  All  above  weirs  were  normal  to  the  flow,  had  vertical  faces  making 
\l/  =  90°  and  had  sharp  crests  consisting  of  a  vertical,  i  inch  planed,  steel  plate 
with  beveled  edge  down  stream.  The  above  quantities  represent  means  of  the 
several  observations  bearing  the  observer's  numbers  in  column  2. 


8o 


HYDRAULICS 


DETERMINATION    OF 


TABLE  III. 

FOR    EQS.    (19)    TO    (22)    WHEN    B  =  b. 


No. 

Original 
Experi- 
ment No. 

Measured  Values. 

Computed 
From  Eq.(22) 

Error 
byEq. 
(6Q) 
Per 
cent. 

B 
m. 

b 

k 

H 

H 

Q 

^s 

M, 

k  +  H 

m. 

m. 

m. 

cu.  m. 

FROM    TABLE    XXm,    EXPERIMENTS    BY    LESBROS,    1829-34.     $= 


33 

i9*f 

o.  202 

O.  2O2 

0.043 

°-°955 

0.68953 

0.012905 

0.4045 

0.6067 

o.o 

34 

i9ff 

O.202 

0.202 

0.048 

o-°955 

0.66551 

0.012561 

0.3987 

0.5981 

+  0.6 

35 

19** 

0.202 

O.  202 

O.OyO 

0.088 

0.55696 

o.  010532 

0.3924 

0.5886 

0.0 

36 

i9fl 

O.2O2 

O.2O2 

O.  IOO 

0.0805 

0.44598 

0.008638 

0.3813 

0.5720 

+0.5 

37 

i9ff 

O.  2O2 

O.202 

0.130 

0.0705 

0.35162 

0.006864 

0.3788 

0.5682 

0.0 

38 

i9f* 

0.202 

O.220 

O.02O 

0.0432 

0.68354 

0.003486 

0-3747 

0.5062 

0.0 

39 

i9H 

O.202 

O.202 

0.030 

0.0378 

0-55752 

0.00244 

0.3380 

0.5069 

o.o 

40 

*W 

O.2O2 

0.2O2 

0.050 

0.0228 

0.31319 

o  .  000843 

0.2653 

0.3979 

+  1.6 

FROM    TABLE   IX,    EXPERIMENTS    BY    BOILEAU,    1845.     ^=90°. 


Eq.  68. 

41 

— 

0.895 

0.895 

0.340 

0.0577 

0.14508 

0.0226025 

0.4031 

o  .  6046 

-3-9 

42 

— 

0.895 

0.895 

0.340 

0.134 

0.2827 

0.0823787 

o  .  4008 

0.6012 

-1.9 

43 

— 

0.895 

0.895 

0.340 

0.219 

0.3918 

0.17702 

0.4029 

0.6042 

-0.6 

44 

— 

1.616 

1.616 

0.468 

0.0937 

0.1668 

0.0861429 

0.4070 

0.6105 

-!-3 

45 

~ 

1.616 

1.616 

0.468 

O.IIO 

0.1903 

o.  108462 

0.4013 

0.6020 

-2-3 

FROM    EXPERIMENTS    BY    J.    B.    FRANCIS    ON    AN    OVERFALL    WITH    CREST 
2.95    FEET    WIDE    AND    FACE   INCLINED.     ^=20°.  DIMENSIONS   IN   FEET. 


B 

b 

k 

H 

V 

Q 

I* 

I* 

p 

46 

89 

9-995 

9-995 

5.048 

0.5872 

0.238 

13-385 

0.3610 

0.4078 

0.8852 

47 

9° 

9-995 

9-995 

5.048 

0.7904 

0.358 

20.892 

0.3581 

0.4062 

0.8816 

48 

9i 

9-995 

9-995 

5.048 

0.9767 

0.480 

28.914 

0.3584 

0.4052 

0.8845 

4Q 

92 

9-995 

9-955 

5.048 

i-3252 

0.725 

46.183 

0-3583 

0.4041 

0.8866 

50 

93 

9-995 

9-995 

5.048 

1-6338 

0.963 

64.346 

0.3609 

0.4034 

o  .  8946 

NOTE.  —  Experiments  41  to  45  were  on  sharp  crested  weirs  and  belong  in  all 
respects  to  the  class  of  experiments  in  Table  II,  though  made  in  metric  measures. 

The  experiments  33  to  40,  while  in  all  other  respects  like  those  in  Table  II, 
were  made  on  very  small  flumes  and  for  weirs  of  very  small  heights  k,  for  which 
Eqs.  (67)  and  (68)  did  not  apply. 

Experiments  46  to  50  were  made  on  a  wide  crested  weir  and  with  ^  =  20°  and 
0=  90°  otherwise  the  weir  dimensions  are  like  those  of  experiments  i  to  3.  Hence 
this  last  set  was  used  to  show  the  effect  of  wide  crests  and  the  values  §  fis  were 
found  from  Eq.  (67)  for  sharp  crested  weirs,  while  the  values  f  /i  were  computed 
from  the  experiments,  using  Eq.  (20).  The  coefficients  p  were  then  obtained 
from  fa  and  ^,  and  the  values  here  found  should  be  compared  with  those  given 
at  the  end  of  this  chapter. 


EMPIRIC  COEFFICIENTS  8  1 

To  reduce  Eq.  (69)  to  feet  units,  the  two  last  terms  in  each  equa- 
tion must  be  divided  by  3.281. 

A  few  experiments  on  a  wide  crested  weir  were  made  by  Francis 
on  the  same  flume  used  for  the  experiments  i  to  3,  and  this  furnished 
a  means  of  determining  p  for  this  particular  case.  The  results  are 
given  in  Table  III,  No.  46  to  50,  see  also  note  at  foot  of  this 
table. 

The  subject  of  weir  crests  and  their  effect  on  discharge  is  of  such 
vital  importance  that  it  will  be  treated  separately;  suffice  it  to  say 
here,  that  the  discharge  for  different  crests  may  vary  20  per  cent 
either  way  from  the  values  obtained  for  standard  sharp  crested 
weirs. 

The  following  Table  IV  contains  results  taken  from  the  Cornell 
University  experiments,  made  for  the  United  States  Board  of 
Engineers  on  Deep  Waterways,  in  June,  1899,  by  Mr.  George  W. 
Rafter. 

These  experiments  cover  a  wide  range  of  head  H,  though  B  =  b 
and  k  are  constant  throughout.  The  values  §^s,  determined  from 
the  new  formulae  (22),  are  seen  to  be  uniformly  decreasing  for 
increasing  values  of  H,  while  Mr.  Rafter's  coefficients,  obtained 
from  Bazin's  formula  Q  =  mbH  \/2  gH,  show  no  regular  law. 

The  new  coefficients  thus  found  were  used  to  determine  the 
constants  in  Eq.  (66)  for  p  =  i,  and  the  following  equation  was  thus 
obtained  for  dimensions  in  feet  : 

|f*.=  0.3851  -  0.0258  H  +  k  +  2jj*  +  0.000132  b  .  (yea) 
and  for  dimensions  in  meters: 


3 

A  careful  examination  of  the  results  in  Table  IV  will  show  a  very 
close  agreement  with  the  coefficients  obtained  in  the  previous  and 
it  should  be  noted  that  the  same  law  of  increase  in  /*s  for  decrease 
in  H  is  clearly  demonstrated  in  all  the  experiments  here  given. 


82 


HYDRAULICS 


TABLE  IV. 

CORNELL    UNIVERSITY    EXPERIMENTS    NOS.     20    AND    21,    JUNE    1899,    G.    W. 
RAFTER.     STANDARD    SHARP    CRESTED    WEIRS    WITH    FREE    OVERFALL. 


Measured  Values. 

Computed  from 
Eqs.  (22). 

Com- 

puted . 

_        t 

No. 

from 

Coef.  m. 

B=b 

k 

H 

V 

Q 

2 

Eq.  (70) 

ft. 

ft. 

ft. 

ft. 

cu.  ft. 

Ps 

IPs 

I 

6-53 

5.26 

o-7 

0.327 

12  -7335 

o  .  4066 

0.5421 

0.4171 

0.4204 

2 

I.O 

0.527 

21-8755 

o  .  4060 

0-5413 

0.4058 

0.4174 

3 

1.5 

0.902 

39-8333 

0.3981 

0.5308 

0.3962 

0.4136 

4 

2.0 

1.283 

60.8596 

0.3912 

0.5216 

0.3909 

0.4106 

5 

2-5 

1.679 

85.0859 

0.3879 

0.5172 

0-3873 

0.4094 

6 

3-o 

2.  062 

111.2059 

0.3831 

0.5108 

0.3846 

0.4094 

7 

3-5 

2-459 

140.  6562 

0.3819 

0.5092 

0.3825 

0.4099 

8 

4.0 

2.851 

172.3920 

0.3806 

0-5075 

0.3808 

0.4112 

9 

4-5 

3.240 

206.4786 

0.3798 

0.5064 

0-3794 

0.4125 

10 

5-° 

3-6I5 

242.1977 

0.3784 

0.5045 

0.3782 

0.4133 

ii 

5-5 

3-979 

279-5493 

0.3761 

°-5OI5 

o-377i 

0.4135 

12 

6.0 

4-335 

3l8-7293 

0.3756 

0.5008 

0.3762 

0.4136 

NOTE.  —  The  above  experiments  were  made  on  a  standard  sharp  crested  weir 
with  0  =  $  =  90°  and  p  =  i. 

(b.)  Coefficients  p>8  for  Eqs.  (19)  to  (22)  for  weirs,  normal  to  the 
channel  but  contracted  by  wing  walls,  for  which  B>b. 

The  experiments  given  in  Table  V  were  used  in  Eqs.  (21)  to  find 
§  i*s  for  the  cases  when  B>b,  and  the  values  thus  found  were 
entered  in  columns  8  and  9. 

The  original  equation  (66)  must  now  be  modified  to  include 
variations  between  B  and  b,  and  may  be  expressed  thus  : 


However,  it  was  found  for  the  experiments  available  for  this  case, 
that  this  equation  did  not  give  more  accurate  results  than  when 
the  factor  of  /?  was  taken  as  b/B,  and  therefore  this  latter  value 
was  finally  adopted. 


EMPIRIC  COEFFICIENTS 


TABLE   V. 
FOR    EQS.  (19)     TO    (21)    WHEN    B>b. 


Measured  Values. 

Computed  from 
Eqs.  (21). 

Errors 

Original 

by  Eq. 

No. 

Experi- 
ment 

B 

b 

k 

H 

Q 

(72) 

Per 

No. 

§**. 

ft 

cent. 

ft. 

ft. 

ft. 

ft. 

cu.  ft. 

FROM   TABLE    XIII,    EXPERIMENTS    BY    J.    B.    FRANCIS,    1852. 


51 

72-78 

13.96 

9-997 

5.048 

0.62355 

16.2148 

0.4047 

0.6072 

—  o.  i 

52 

56-61 

13.96 

9-997 

5.048 

0.79899 

23-4305 

0.4014 

0.6021 

o.o 

53 

n-33 

13.96 

9-997 

5.048 

0.99732 

32.5798 

0.3992 

0.5988 

o.o 

54 

5-10 

13.96 

9-997 

5.048 

i-24757 

45-5654 

o-3979 

0.5969 

o.o 

55 

1-4 

13.96 

9-997 

5.048 

i  •  55079 

62.6019 

0.3929 

0.5894 

+  1.0 

56 

79-84 

13.96 

9-997 

2.014 

0.64928 

17.4428 

0.4023 

0.6034 

+0.3 

57 

62-66 

13.96 

9-997 

2.014 

0.82624 

25.0410 

0.4000 

0.6000 

+  0.2 

58 

36-43 

13.96 

9-997 

2.014 

1-05033 

36.0017 

0.3989 

0-5983 

o.o 

FROM    TABLE    XXVUI,    EXPERIMENTS    BY    FTELEY    AND    STEARNS,    1878. 


59 

4 

5-° 

3.008 

3-56 

0-2155 

1.007 

0.4146 

0.6220 

+  1.0 

60 

9 

5-° 

3.0081 

3-56 

0.3301 

1.869 

0.4062 

0.6092 

o.o 

61 

14 

5-° 

3.008 

3-56 

0.4843 

3.284 

0.3990 

0-5985 

—  O.I 

62 

31 

5-° 

3.007 

3-56 

0.7398 

6.134 

0.3930 

0.5894 

o.o 

63 

40  &  45 

5-o 

3.0101 

3-56 

0.8708 

7.8075 

0.3904 

0.5856 

+  0.2 

64 

16 

5-° 

2.3132 

3-56 

0.5824 

3.284 

0.3941 

0.5911 

-0.5 

65 

24 

5-° 

2.3125 

3-56 

0.7478 

4.736 

0.3899 

0.5848 

—  O.2 

66 

35 

5-o 

2.3126 

3-56 

0.9548 

6.796 

0.3867 

o.  5801 

O.O 

67 

!9 

5-° 

4.0058 

3-56 

0.4978 

4.636 

0.4038 

0.6057 

O.O 

68 

38  &39 

5-o 

4.007 

3-56 

0.70595 

7-8i5 

0-3974 

0.5916 

+  0-4 

69 

20 

5-° 

3-3io7 

3.56 

0.5678 

4-637 

0.4018 

0.6027 

-0.9 

70 

32 

5-° 

3-3ioi 

3-56 

0.6860 

6.134 

0-3993 

0.5989 

-0.9 

7i 

48  &49 

5-o 

3-3095 

3-56 

0.9307 

9.6485 

0-3956 

0.5934 

-0.7 

NOTE.  —  All  of  the  above  experiments  were  made  on  sharp  crested  weirs  placed 
normal  to  the  flow  and  vertical,  so  that  <f>  =  i/'  =  90°.  While  B  >  b,  this  differ- 
ence was  equally  divided  on  both  sides  for  all  experiments  51  to  66,  though  for 
the  five  last  lines,  67  to  71,  the  channel  contraction  was  all  on  one  side  of  the 
flume. 


84 


HYDRAULICS 


The  constants  were  thus  determined  from  the  experimental 
values  51  to  66  and  furnished  the  following  for  dimensions  in  feet: 

0.007328  .    x 

-—~  ~  +0.00093  b  -  (72) 


z.      /     \ 
+0.00305  b  .  (73) 


2  /  b 

-  K  =  0.3655  +  0.02357  ^- 

and  for  dimensions  in  meters: 

2  /b\      0.002384 

-  /i.  =  0.3655  +  0.02357^  +  --  g^  - 

In  the  same  table  all  the  values  for  f  /*,  were  recomputed  by  Eq. 
(72)  and  the  resulting  percentage  errors  entered  in  the  last  column. 
This  gave  very  concordant  results. 

However,  for  weirs  in  which  k  becomes  an  important  factor,  it 
may  eventually  be  necessary  to  employ  the  more  complicated 
form  (71). 

For  weirs  built  diagonally  to  the  flow,  the  above  coefficients 
must  necessarily  answer  the  purpose  until  new  experiments  on 
such  weirs  shall  have  been  made. 

3.    INCOMPLETE    OVERFALLS. 

Very  few  experiments  have  ever  been  made  on  incomplete  over- 
falls. In  these  the  discharge  takes  place  partly  through  a  sub- 
merged area  and  partly  as  into  free  air,  and  coefficients  i^l  and  /* 
have  been  assigned  to  the  two  discharges  respectively. 

Strangely  enough,  the  submerged  section  has  usually  received 


Fig.  31 

the  larger  /*  and  as  this  seems  contrary  to  all  reason  the  following 
explanation  is  offered. 

Let  Fig.  31  represent  the  discharge  section  in  which  A  BCD  is 
the  actual  opening  and  AecdjB  is  the  contracted  discharge  section 


EMPIRIC  COEFFICIENTS  85 

as  for  discharge  into  free  air,  making  ft  =  0.72.  Now  if  the  lower 
portion  of  the  section,  EFCD,  is  submerged,  then  most  of  the  con- 
traction belongs  to  this  portion,  while  the  upper  area  AEFB 
includes  little  of  the  contracted  area.  If  these  areas  retain  any- 
thing like  this  relation  after  the  lower  portion  of  the  section  is  sub- 
merged, then  certainly  the  ratios  which  the  contracted  areas  bear 
to  the  respective  flow  areas  must  be  less  for  the  submerged  portion. 
Hence  it  will  be  natural  to  assume  ^  <  ft,  and  as  there  is  great 
liability  of  meeting  with  unforeseen  influences  on  a  partially  sub- 
merged section,  the  former  values  ^  for  complete  overfalls  cannot 
safely  be  employed  here. 

The  experiments  of  Francis,  given  in  Table  VI,  offer  a  valuable 
contribution  to  the  present  subject  and  from  them  values  of  /^  and 
ft  were  determined  for  Eqs.  (28). 

The  final  equation  (28)  when  solved  for  /^  or  ft  gives 


Q-^b\H1-^\Vg(Sl  +  S2) 

.    .     (740) 


(74&) 


&  \H.  - 


either  of  which  may  be  used  when  all  the  other  terms  are  deter- 
mined from  experimental  data,  but  a  single  set  of  observations 
will  not  be  sufficient  to  solve  one  equation  with  two  unknowns. 

The  following  process  was  employed  for  the  determination  of 
ftx  and  ft. 

Values  were  substituted  in  Eq.  (740)  for  two  sets  of  observa- 
tions having  approximately  the  same  dimensions  for  Hy  Now 
from  the  behavior  of  f  ft  for  complete  overfalls,  it  is  found  that 
for  similar  values  of  H2,  f  ft  is  practically  the  same.  Hence,  each 
such  pair  of  equations  may  be  equated  so  as  to  involve  only  filt 
which  would  then  represent  the  mean  value  for  the  two  sets  of 
observations.  Then  using  this  value  of  ft1  in  each  of  the  pairs 
of  equations,  the  mean  value  f  ft  was  found. 


86 


HYDRAULICS 


CQ 


.  o 


<y  < 

j   w  s 


d     ddddddddddd 

+      +  +  I    I    I    I    I  +  + 


•2  3 


"S  W  1  W 


M     Tj-00 


cooo 


r-     O   ^t 
+      I     I 


N    10  H  00  00 
M       VO     CO  « 

I    11  + 


<N       ro\O    t^ 


«Si  00    M    rf  CM    O 


iOMOOOMON  er>00 
NMOONON^^J-oOt^ 

CSCNMHMHMOO 


O 
OO 


t^-OO 
NO  O 


OOOOOOOOOOO 


ddoooooo 


O         OOOOOOMMOHH 


ON 

^"  N 


^-  loO    t^1*  O^  O^  w 

d  o  d  d  o  o  w 


ON     t^O  OO 
CO     vo 
O     ^ 


t^.   W     H  NO   OO     t^- 
O     ^~   M   O     M     N 

M      Q    Q    OOOOWHOOO 


M       ON  ON  ON  ON  N    O)    t^OO    -*x 


js 

• 


Hi 


bfl  *«3  ^"^ 

fi       g  W 

"frt      >-(  O^ 

o     O  pq 

^•5  . 


C  o     w 

nJ  <u     ., 

J  {3    ^ 

O  Q,  (u 

^3  as   ^ 


-a  I 

^  ^   53 

#1 

H  2 


EMPIRIC  COEFFICIENTS  87 

Now  Eq.  (67)  permits  of  finding  very  close  values  for  the  differ- 
ences between  successive  values  §  /*  when  the  corresponding 
differences  in  H2  are  known.  Hence  the  first  values  of  J  /*,  above 
found,  were  corrected  for  these  small  differences  and  the  cor- 
rected values  then  used  to  determine  the  final  values  of  /^  from 
Eq.  (746).  These  latter  values  are  entered  in  columns  14  and  15 
of  Table  VI. 

Using  these  coefficients  in  Eqs.  (28)  the  quantity  was  computed 
for  each  of  the  tabulated  experiments  and  these  results  entered  in 
column  20.  The  remarkably  close  agreement  which  these  quanti- 
ties bear  to  the  measured  values  is  shown  by  the  percentage  differ- 
ences in  column  21. 

For  the  sake  of  comparison,  the  values  //,  from  Eq.  (67)  were 
computed  for  each  experiment  and  entered  in  column  13,  and  the 
percentage  differences  between  these  and  the  //'s  in  column  14 
are  given  in  column  16,  while  the  percentage  differences  between 
each  pair  of  /*t  and  //  are  entered  in  column  15. 

In  column  19  the  values  of  Q  are  given.  These  result  from  the 
Francis  formula  (470)  as  follows: 

Q  -  3.33  b  (H  -  H,)»  +  4.5988  bH  VH-H,.    .  (75) 

TT 

The  values    -^    were  also  computed  as  given  in  column  18, 
Jn. 

because  the  coefficient  C  in  the  formula  of  Messrs.  Fteley  and 
Stearns  is  made  to  depend  on  this  ratio. 

By  careful  inspection  of  Table  VI  the  following  conclusions 
may  be  drawn: 

TT 

1.  The  ratio  — 1  does  not  seem  to  have  any  definite  relation  to 

JdL 

either  of  the  coefficients  /*  except  that  for  all  ratios  exceeding  0.5 
the  values  /*  are  irregular. 

2.  This  irregularity  in  the  coefficients  for  the  first  three  experi- 
ments, 25,  95  and  96,  for  which  H2  was  small  as  against  the  sub- 
merged depth  Hv  is  probably  due  to  the  very  large  value  of  k 
compared  with  both  Hl  and  H2.     It  is  possible  under  these 


88  HYDRAULICS 

extreme  conditions  that  the  impact,  of  the  flow  of  approach  against 
the  weir,  exerts  an  influence  on  the  whole  discharge  area. 

3.  The  values  /*,  in  column  14,  increase  rapidly  for  the  first 
four  experiments  and  then  slowly  diminish  with  increasing  values 
of  H2  as  is  indicated  by  the  percentages  in  column  16. 

4.  The  coefficients  JJLV  in  column  15,  follow  an  exactly  opposite 
law  of  increase,  and  from  the  fourth  experiment  down  ^  is  from 
15.5  to  12  per  cent  less  than  the  corresponding  p. 

5.  Francis  observed  the  fact  that  for  small  values  of  Hl  (from 
0.08  to  0.17  feet)  the  discharge  over  the  crest  takes  place  as  for 
discharge  into  free  air.     That  is,  some  air  is  still  carried  over  the 
crest  and  no  suction  occurs  on  the  discharge  area.     When  Hl 
becomes  larger,  this  air  is  expelled  and  suction  effect  follows  with 
increased  discharge.     Hence  it  is  very  important  in  conducting 
experiments  of  this  kind   to  note  these  conditions  carefully  lest 
the  entire  work  may  prove  useless. 

6.  The  present  experiments  do  not  manifest  the  same  accuracy 
as  was  observed  for  those  dealing  with  complete  overfalls.     Also 
the  quantity  of  discharge  was  not  directly  measured  but  computed 
from  velocity  experiments. 

Hence  there  is  still  need  for  further  experiments  on  incomplete 
overfalls  under  more  varying  conditions  as  when  B  >  0,  etc., 
and  careful  attention  should  be  given  to  measure  such  dimensions 
as  will  make  possible  the  separate  determination  of  /^  and  //. 

It  was  impossible  to  fit  any  formula  to  the  values  /^  and  /*  of 
the  first  three  experiments.     For  heights  H2,  between  0.643 
1.119  feet>  tne  following  formulae  were  adopted: 

For  dimensions  in  feet: 

.        O.OI038        .  ,     T 

/*  =  0.4001  +  — 77^-  +  0.000146  0 

.     .  (760) 

jMt=  0.5274   +  O.OOOI46  0, 

and  for  dimensions  in  meters: 

0.00316 


£j  =  0.5274  H-  0.00048  b. 


f  0.00048  b        _(766) 


EMPIRIC  COEFFICIENTS  89 

When  the  pressure  height  H2  becomes  larger,  the  following  for- 
mulae are  preferable: 
For  dimensions  in  feet: 

f  /JL  =  0.4001  +  a°°799  +  0.000146  b 
£LO 


(77*) 


Hi==  0.5346  +  0.000146  &, 

and  for  dimensions  in  meters: 

o                       ,  0.00244  0  , 

4  IJL  =  0.4001  H -f  0.00048  o 

^2 

^i==  0.5346  +  0.00048  ft. 

All  the  above  formulae  (76)  and  (77)  are  for  sharp  crested  weirs 
for  which  p  =  i.  For  any  other  kind  of  crest  the  values  for  p 
must  be  determined  from  experiments  still  remaining  to  be  made. 

4.    SLUICE   WEIRS   AND    GATES. 

Among  the  available  experiments  which  can  be  used  for  the 
determination  of  JJL  in  Eqs.  (36),  there  are  only  a  few  which  are 
entitled  to  any  confidence.  Those  given  in  Table  VII  comprise 
about  all  for  which  sufficient  data  were  given  to  permit  of  their 
use. 

The  largest  experiments  of  this  class  are  those  made  on  the 
Danube  Canal,  1876  to  1883,  by  Freiherr  v.  Engerth.  These  three 
experiments  are  tabulated  as  Nos.  i,  2  and  3  and  the  /^  were 
found  by  inserting  the  observed  values  in  the  new  formulae  (36), 
using  n  =  0.67. 

For  further  values  /^  the  experiments  given  in  the  table  were 
used.  These  were  all  made  prior  to  1880,  by  the  following: 
No.  4-11  by  Lesbros;  No.  12-22  by  Boileau;  No.  23-34  by 
Weisbach;  and  No.  35-41  by  Bornemann.  The  computation  of 
values  /^  for  these  experiments  in  Eqs.  (36)  resulted  in  figures 
entered  in  the  table,  and  the  percentage  differences  between 
these  and  the  values  found  from  Eqs.  (78)  to  (81)  are  given  in  the 
last  column. 


90 


HYDRAULICS 


/"""s 

«illl^ 
s   -§£^ 

Q       «       j> 

M  M~M 

iw^ 

-^ 

«°S 

*i| 

£ 

O         t^        co 
0          <N         O 
•+        + 

MM^MVOOOt^ 
OCOMOMWON 

+  +      1        1        1        1               + 

W     M     M  VO     TfvO     •*    O 

esiMMc^OOMO 

+++  1  +  1   1 

1    1    1 

a^S-sf 
*«^1 

s!3 

oo        vo       vo 
00       CO       do 
odd 

VOVO-^MOO    lOQsVO 
covocSi-icO    vovo^J- 
O\t^t^-t^-cocOM    O\ 
vOvOO-OOvO-O    vo 

dodddddd 

<S     M     ONvNOO     COMt>- 

00    ONt^QvOOO    ONI^ 
VO  VO  VOVO    VO  VO  VO  VO 

dodddddd 

fIS 

odd 

01 

if* 

^     vo       *^ 

S      S      « 
odd 

co  O  co  voO    t^.00    co 
rf  vo  O    M    O  O    t^cd 
ON  t-»OO    t^  ^-  CO  M    O\ 
vOvOOO^OOO    vo 

M    O\vO    ON  CO  W  00  vO 
OS  ONOO    O    t»  ON  ONOO 
\O  vo  vovO    vo  vo  vo  vo 

vo  t^  10 

00      t^.    M 

vo  vO    t» 

oooooooo 

OOOOOOOO 

O   O   O 

Measured  Values. 
All  in  Meters. 

O* 

ON        co        co 
0*            !>.          CO 

\O    t^  O\OO    co  M   vo  t^ 

M     **»  VOOO      M     O   ^    M 

M   COCOM  VO«N   t^»vo 
t^HC4MO\OOcO 

vo  t^  co  t^vO          O 
vo  cooo    t^.  (S  00    O    "*• 
ON  N  vO    t^GO    Tt-  t^  Tf 
t^O    «M    t^vocowvo 

coO    co 

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vO    co  ON 
00    ON  ON 
O    0    0 

O          •**         t-~ 

O    O    O    M    N    ^j-O  OO 

oooooooo 

OOOOOOOO 

000 

nf 

t^     <+     2 

ON            HI              Tj- 
O              M              M 

CO 

1OO     MO     MWt^-O 

vo  t^  O    t^OO    l^-  co  ON 
<N    t^  VOOO  O  00     O  00 

IO                         VO 

VOCOOMCOVOO-<t 

ON  O    O    Tf  Tfvo    co  Tf 

M     VO   ^   VO   M     COO     TJ- 

vo  O    vo 

oo  oo  M 

ON  M    «N 

O      M      M 

OMOMOHHM 

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0    0    O 

H~ 

O         t^       00 

M              CO            <N 

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vo  vo  v/-. 

O      M     CO 

ON  M    M 

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oo"          f^       00 

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ON  ON  10 
00    CM    Tt 
VOVO    VO 

odd 

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0          0          O 

odd 

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dodddddd 

o  o  o 
odd 

VOVOVOVOVOVOVOVO 

dodddddd 

d 

to        w         \r> 

M         ro        o» 

CO           W             CO 

cocovovoO    O    O    O 
OOOOttW^J--<t 

dodddddd 

vo  vo          n-  rj-  1» 

CX300    O    O    ONONO\O 
•*  -*00  00    ON  ON  ON  W 
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dodddddd 

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VO          to          10 

o      vd      o 

Tf          rj-          rl- 

vOvOvOvOvO\OvOvO 

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00  00                     CO 
ON  ONOO  00    ON  ON  ONOO 

dodddddd 

o  o  o 

ON  ON  ON 

o*  d  d 

« 

0        *0          0 

odd 

vo       to       10 

S^l^S^^^^ 

CO  co  co  co  co  co  co  co 

00  00                     00 
ON  ONOO  OO    O\  ON  ONOO 

dodddddd 

000 

ON  ON  ON 

odd 

•OM  I*0!3!-1© 

.  vO      •    1-1          co 
N    t>.  COOO       •  00 

^xCO  \oo   M  oo 

OwO    co  M  00  vO    CO  M 

CO   M    10  Tj-    O     t^-VQ     M 
M                       M 

* 

1    1    ' 

N            •** 

(N                M               VO 

a 
M 

H            N            CO 

•*  vovO   t^-00    ON  O    M 

N    co  rl-  vovO    f»00    ON 

§  S  S 

Iq  juauiuadxg  |        -i^jaSug 

•sojqsai 

•nTOipg 

8 


EMPIRIC  COEFFICIENTS 


*•£§ 

* 

£S% 

xo  Tf  to  04    O    t"- 

ft-   CO  M     M     O     C*    *•» 

O   O   O   O   M   M 

1    +    1     1     1    + 

1     1    1    1     I     1 

co  t^  W  vo  VO   l*»  O 

+  +  1  1  1  1  1 

co  ***•  O    co  ON  O 
r>-  t»oo  oo  O  t-~ 

04    Tf  00  00    O    O 
J>.  t^O   t»»  t^oO 

XOVO    XO  XO  0*    t>.  ON 
t^vO   t^  t^OO  vO  O 

000000 

O    O   O   0   O   O 

O    0    O   O   O   O   O 

O  vO    Tf  co  ON  xo 
IO  t^-  04    XO  O    O4 

t-»  t^oo  oo  t^>  «*• 

00    Tf  to  Tf  M  vo 
to  ON  M    Tf  Tf  O 
t"^  t^»  t^^OO   t^OO 

w   xo  co  •*  ON  N   co 

•*    O     M     ON   M     M     tH 

oo  t^oo  *>.  ON  t^  r- 

0    O    O   O   O   O 

o  o  o  o  o  o 

O   O    O    0   O   O   O 

r»oO    04   O  to  04 

vO  00    xo  co  ON  O 

rj-  to  <N  r^oo  vO 

N     ONOO     M    XO  t^- 
Tt   ON  XO  M     CO  T*1 

t-00    O    0  VO    PO 

M     M     M     CO  N     CO 

0    O    O    O    O    O 

t^  t^  ON  ON  ON 
xo  to  t^  t^.  rf  Tt-  ^~ 

O   O    0    O   O   O 

666666 

0    O    O   O   O   O   O 

xo  M        O    Tf  04 
ON  O   Is*  M    O    M 

Tf  ON  CO  ON  O  00 

M     <N     M     O     CS     O) 

w   co  O   «  tooO 

VO     O     O  00     ON   M 

xovo  ON  r^-  1^-  ON 
0    0    0    0    O    O 

Tj"    ON   ON  t^^*  M     M     O4 

M  POOO  oo  M  co  to 
O   O    O    M   O    O    M 

O   O   0   O   0    O 

o  o  o  o  o  o 

O   0    O    O   O    O   O 

O  OO  vO    M  vO    O 
rfvo    M    POO    rf 

04     tO  O     O     CO  Tf    Tf 

vo    TT  ON  ON  w  OO 

^t    f^    S^    M      N      fSl~    S 

M     M     M     M     <N     M 

r*j  r*;  n    cs    cs    cs    CN 

0    O   O    O   O   O 

0    O    O    O    O    O    O 

:::::: 

W    M  vO    PO  M  OO 
O     f*.   M     M  VO     XO 

CM     O  OO    *>•  O    **» 

O    T^-  ON  t^  •"*•  xoO 
xo  ON  t>-  ^  t^-  O    xo 
CO  co  CO  TJ-  04    CO  co 

O    0   0    O    O    O 

O    O    O   O   O   O   O 

O    O    O    O   0    0 

666666 

O    0    0    0    0    O 

666666 

0    0    O    O    0    0    0 

6666666 

eg\g  s-  £•«•«- 

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XOXO  XO 

to  xo  xo  f"*-  ^*»  t^*» 
0    0    0    O    0    0 

T}-OO     t>»   M     M     O     M 
O     XO  04     ON  PO   M    XO 
O4     M     M     O     M     M     O 

0    0    0    O    O    0 

O    O    O    0    0   O 

O    O   O    O   O   O   O 

xo  xo 

ON  ON              xo  xo 

CO  CO  CO  <M    M    <N 
PO  CO  PO  CO  CO  PO 

O    O    O    O    rj-  «j-  Tf 

O4     04     O4     04     t^-  t^  t^ 

xo  to  xo  xo  t^  Is*  t>- 

000000 

0    0    0    0    0    O 

O    O    0    0    0    0    0 

xo  xo 

ON   ON                    tO  XO 

666666 

CO  PO  CO  PO  CO  PO 

666666 

Tf    Tf    Tf    Tf    W      04      04 

Tf    Tf    Tf    Tf    O      O      O 

to  xo  xo  XOCO  OO  OO 

6666666 

M     M 

*      * 

POOO    Tf  O   ON  w 

M              M 

M  VO    ON  XO  t~-  ^  04 

MM                      M 

PO  Tf   XOVO     t^OO 

ON  O    M    M    CO  rj- 

toO   f»00    ON  O    M 
CO  co  co  CO  CO  Tf  Tf 

•qow 

ISP.&. 

•uumuaujog 

III 

g  1 1  -a 

^^^^ 


92  HYDRAULICS 

The  coefficients  /*,  which  the  various  observers  found  from  the 
use  of  their  own  formulae,  are  given  in  the  i2th  column,  and  the 
following  comments  regarding  these  may  not  be  out  of  place  here. 

Excepting  the  first  three  cases,  the  general  comparison  shows 
the  new  values  to  be  somewhat  less  than  those  given  by  the  observers 
themselves,  and  this  is  undoubtedly  so  because  the  new  formulae 
take  into  account  several  conditions  not  considered  by  the  older 
formulae. 

According  to  the  experiments  of  Engerth,  Boileau  and  Weis- 
bach  on  sluices  wherein  the  sill  was  level  with  the  bottom  of  the 
flume  or  canal,  or  when  k  =  o,  H2  varies  with  pv  while  accord- 
ing to  Lesbros  for  k  =  0.54  m.  this  relation  is  inverse.  The  only 
experiments  which  form  exceptions  to  this  rule  are  Nos.  2,  21,  31 
and  33,  for  which  reason  it  is  fair  to  assume  that  some  small  errors 
were  committed  in  those  cases. 

Weisbach's  experiments,  23-28,  were  made  on  a  sluice  without 
side  and  bottom  contractions  and  a  rounded  edge  on  the  gate, 
which  explains  the  rather  high  values  there  obtained  for  //r 

It  should  also  be  observed  that  for  these  experiments  the  coeffi- 
cient jt£j,  for  submerged  discharge,  is  smaller  than  for  free  dis- 
charge, which  is  exactly  the  opposite  of  our  previous  findings  in 
this  regard.  However,  the  present  case  is  different  inasmuch 
as  the  contraction,  whether  for  free  or  submerged  discharge,  takes 
place  all  around  the  opening  when  dealing  with  a  sluice  gate. 
Also  the  suction  produced  by  the  water  leaving  the  discharge 
section,  tends  to  increase  the  quantity  for  submerged  discharge 
as  against  free  discharge  where  no  suction  occurs.  Therefore,  these 
phenomena  would  indicate  just  what  the  values  //  indicate  to  be 
the  true  condition. 

Hence  these  tabulations  of  /^  and  /*  show  quite  conclusively 
that  the  new  formulae  apply  to  a  rather  wide  range  of  conditions 
and  can,  therefore,  be  expected  to  give  very  much  greater  accuracy 
than  was  possible  to  attain  with  the  older  formulae  which  did  not 
include  the  many  variable  influences  always  attending  hydraulic 
problems. 


EMPIRIC  COEFFICIENTS  93 

However,  these  experiments  are  of  such  nature  that  their  prac- 
tical applicability  is  very  limited  and  until  more  extensive  experi- 
ments of  this  kind  shall  have  been  made  these  values  must  be 
accepted  as  the  best  information  available  at  this  time. 

The  following  formulae  for  /*  and  /*t  are  given  with  reserve 
because  the  data  on  which  they  are  based  is  entirely  inadequate 
for  the  deduction  of  such  formulae.  However,  they  are  better  than 
nothing  and  may  occasionally  serve  a  useful  purpose  in  want  of 
something  better. 

Bornemann  gave  a  formula  for  /^  which  seems  to  give  fairly 
accurate  results.  This  formula  with  slightly  modified  coefficients 
and  another  member  involving  b  was  used. 

For  the  first  three  experiments  by  Engerth,  there  is  not  enough 
data  to  justify  any  formula;  however,  the  following  one  is  offered 
for  submerged  discharge  through  large  sluice  openings : 

For  dimensions  in  feet: 

0.2717  \/a 
^  =  0.7069 —  +  0.00093  b  •     •     •    (7&0 

H-- 


and  for  dimensions  in  meters: 

'a 


+  0.00305  b    .     ..    .     (78^) 


The  other  experiments  on  submerged  discharge  are  too  unreliable 
and,  therefore,  formulas  could  not  be  found  except  for  the  last 
experiments  35-41,  by  Bornemann,  for  which  the  following  are 
given  : 

For  dimensions  in  feet  : 

/*!  =  0.4988  +  ^^  —  ?  +  0.00093  &  •    -     • 


H-- 


and  for  dimensions  in  meters: 


0.4988  +  °'149  5  —  2  +  0.00305  b  . 
H-- 

2 


94  HYDRAULICS 

The  experiments  4-11  by  Lesbros,  for  free  discharge  with  com- 
plete contraction,  give  for  dimensions  in  feet  : 

0      0.0135=;  \/a      0.0382  • 

H  =  0.5708  H  --  OJJ         +  —  ^z:  +  0.00131  b     .     (8oa) 


and  for  dimensions  in  meters  : 

0.01355  *a    i    0.021          ,  ,  ,_    ,x 

/£  =0.5708  +   —  ,  °°°         H  --  ;=-*  +  0.00431  b    .    (806) 


For  the  experiments  12-19  by  Boileau  on  free  discharge  through 
gates  of  same  size  as  the  flume,  without  bottom  sills,  thus  allowing 
the  discharge  to  proceed  without  side  or  bottom  contraction,  the 
formula  becomes  for  dimensions  in  feet: 


0.004Q2     ,  ,  , 

-i 3*-  +  0.000146  b  . 

a 


2 
and  for  dimensions  in  meters: 


0.0l8o8  Aa         0.00144     ,  or  /o    I.\ 

-575  !  --  /••  +  *"   —     +  0.00048  6    .    (810) 


For  experiments  23-28  by  Weisbach,  for  free  discharge  like 
experiments  12-19,  without  contraction  but  having  the  bottom 
edge  of  the  gate  rounded  off,  the  following  formula  was  found  for 
dimensions  in  feet: 

0.21036  \/a      0.00718  ,.  ,      /o\ 

//  =  0.8452  --  1°  H  --  '-  —    +  0.000146  b  .  (820) 


and  for  dimensions  in  meters: 

0.21036  Va      0.00210  ,  0,         /0  ,x 

H  =  0.8452  --  ^  -i  ---  -  +  0.00048  b    .    (826) 


EMPIRIC  COEFFICIENTS  95 

5.    END  CONTRACTION. 

In  all  the  new  formulae  it  is  supposed  that  the  discharge  continues 
with  a  uniform  width  of  section  b  which  is  true  when  the  water, 
after  leaving  the  weir  crest,  is  confined  by  lateral  walls.  All  of 
these  cases  are  known  as  discharge  without  end  contraction,  the 
term  end  being  applied  to  the  ends  of  the  weir  adjacent  to  the 
sides  of  the  canal. 

When  the  discharge  is  not  thus  laterally  confined  and  is  allowed 
to  take  place  into  free  air,  then  end  contraction  takes  place  and 
the  quantity  of  discharge  is  slightly  reduced. 

The  exact  amount  of  reduction  thus  produced  can  only  be 
estimated.  Francis  proposed  a  simple  formula  for  this  purpose 
in  the  form  of  a  correction  to  the  length  b  of  the  weir,  by  sub- 
tracting an  amount  o.i  H  for  each  such  end  contraction. 

Thus  the  new  b  =  (b  -  o.  i  nH)     .     .     .     .     .    .    . ...     (83) 

where  n  =  number  of  end  contractions. 

Undoubtedly  this  covers  the  case  in  a  very  approximate  manner 
but  in  the  absence  of  other  more  extended  experiments  to  determine 
the  effect  of  end  contractions  on  the  discharge,  the  above  advice 
must  be  followed. 

6.   WEIR  CRESTS. 
Coefficient  p  for  Different  Styles  of  Crests,  for  Complete  Overfalls. 

The  following  data  have  been  taken  from  pp.  71-75,  "  Hydraulic 
Tables"  by  Professor  Gardner  S.  Williams  and  Mr.  Allen  Hazen, 
published  by  Messrs.  John  Wiley  and  Sons,  in  1905.  For  the 
kind  permission  granted  by  these  gentlemen,  to  use  this  very  val- 
uable collection  of  data,  the  author  here  repeats  his  expression  of 
gratitude  and  thanks. 

The  following  tables  were  derived  from  experiments  conducted 
under  the  personal  supervision  of  Professor  Gardner  S.  Williams 
at  the  hydraulic  laboratory  of  Cornell  University  between  1899  and 
1904.  Some  of  these  experiments  were  made  for  the  United  States 
Geological  Survey,  some  for  the  United  States  Board  of  Engineers 


96  HYDRAULICS 

on  Deep  Waterways,  and  those  on  the  Croton  Dam  model  were 
made  at  the  request  of  Mr.  John  R.  Freeman.  The  figures,  as 
here  given,  represent  the  results  deduced  by  Professor  Williams, 
and  are  entitled  to  the  utmost  confidence. 

The  triangular  weirs  with  sloped  face  upstream,  are  omitted 
here  because  this  effect  is  included  in  the  new  formulae. 

In  general  then,  the  coefficient  p  from  Eq.  (66)  is  found  to  be 

p  =  —  t  where  ft  is  the  empiric  coefficient  for  discharge  over 

standard  sharp  crested  weirs  and  coefficient  /*  is  the  for  any  other 
crest.  Hence,  having  found  ft  from  the  foregoing,  the  /*  for 
any  particular  kind  of  weir  crest,  other  than  the  standard,  may  be 

found  from 

/*  =  ftp,    or   f  p  =  f  ft§p (84) 

This  is  equivalent  to  calling  p  a  multiplier  with  which  to  multi- 
ply ft  for  standard  sharp  crested  weirs,  to  find  /*  for  any  weir 
of  exactly  the  same  general  dimensions  and  conditions  of  flow 
differing  only  in  shape  of  the  crest.  Hence  p  may  also  be  called 
the  crest  coefficient,  which  represents  the  change  in  the  contrac- 
tion of  the  discharge  section  due  to  the  change  in  the  crest  from  the 
standard  sharp  crest. 

In  order  that  p  may  be  determined  experimentally  for  different 
kinds  of  crests  it  is,  therefore,  necessary  to  make  duplicate  experi- 
ments first  with  the  standard  sharp  crest  and  then  with  the 
experimental  crest,  all  other  conditions  remaining  exactly  alike. 

This  was  not  done  by  Mr.  Rafter  for  the  very  elaborate  experi- 
ments made  by  him  at  Cornell  University  in  1899.  There  all  the 
experiments  for  standard  sharp  crested  weirs  were  confined  to 
varying  the  head  H,  all  dimensions  of  weir  and  channel  remaining 
constant.  The  experiments  on  other  forms  were  widely  varied  in 

height  k  and  other  dimensions  and  hence  the  values  —  expressing 

the  relation  intended  to  be  expressed  by  p,  did  not  contribute  much 
of  scientific  value,  because  the  cases  thus  compared  were  not 
strictly  comparable. 


EMPIRIC  COEFFICIENTS  9^ 

The  following  data  from  the  above  named  "Hydraulic  Tables" 
is  believed  to  be  the  most  accurate  in  existence  at  this  time  and 
aside  from  a  few  general  remarks  the  tables  are  self  explanatory. 

All  the  formulae  previously  presented  do  not  include  the  element 
of  weir  crests  and  the  above  values  for  /£,  are  for  sharp  crested 
weirs  of  standard  type,  so  that  the  /*,  once  found  for  a  particular 
standard  weir,  the  multiplier  p  to  be  used  to  find  //  for  any  other 
than  standard  weir,  is  given  below. 

"  On  all  the  models  having  vertical  downstream  faces,  including 
model  P,  air  was  admitted  to  the  space  underneath  the  sheet. 
On  models  D  and  E,  experiments  were  made  with  the  space  under- 
neath the  sheet  unaerated,  so  that  a  partial  vacuum  existed  there, 
which  is  shown  to  increase  the  discharge  about  5  per  cent  at  the 
high  heads.  For  the  weirs  with  inclined  downstream  faces,  models 
F  to  O  inclusive,  no  air  was  admitted  under  the  sheet.  A  com- 
parison of  the  results  upon  models  G  and  H,  shows  the  effect  of 
rounding  the  upstream  corner  of  the  weir  to  be  an  increase  in  dis- 
charge of  about  4  per  cent  at  the  high  heads." 


98 


HYDRAULICS 


TABLE    VIII.  —  RECTANGULAR  FLAT  CRESTED  WEIRS. 
VALUES    OF    p    FOR    SAME    6  AND  k. 


H 
Feet. 

When    w  = 

0.48  ft. 

0.93  ft. 

1.65  ft. 

3.i7ft. 

5.84  ft. 

8.98  ft. 

12.24  ft. 

16.30  ft. 

o-5 

0.902 

0.830 

0.819 

0.797 

0.785 

0.783 

0.783 

0.783 

I.O 

0.972 

0.904 

0.879 

0.812 

0.800 

0.798 

o-795 

0.792 

i-5 

I  .000 

0-957 

0.910 

0.821 

0.807 

0.803 

0.802 

0.797 

2.O 

I.  000 

0.989 

0.925 

0.821 

0.805 

0.800 

0.798 

o-795 

2-5 

I.OOO 

I.OOO 

0.932 

0.816 

0.800 

o-795 

0.792 

0.789 

3-° 

I.  000 

I.OOO 

0.938 

0.813 

0.796 

0.791 

0.787 

0.784 

3-5 

I.OOO 

I.OOO 

0.942 

0.810 

°-793 

0.787 

0.783 

0.780 

4.0 

I  .000 

I.OOO 

0.947 

0.808 

0.790 

0.783 

0.780 

0-777 

H  =  depth  of  water  flowing  over  the  crest  of  the  weir. 

TABLE    IX.— COMPOUND    WEIRS. 
VALUES  OF  p  FOR  VARIOUS  TYPES  OF  WEIRS  F  TO  L. 


H 
Feet. 

Type  F. 

Type  G. 

TypeH. 

Type  I. 

Type  J. 

Type  K. 

TypeL. 

o-5 

0.964 

0.932 

0-934 

0.968 

0.971 

0.971 

0.971 

I  .0 

1.026 

0.982 

I  .000 

1.008 

1  .040 

1  .040 

0.983 

!-5 

1.064 

1.015 

1  .040 

1.032 

1.083 

1.092 

I.  012 

2.0 

1.  066 

1.031 

1.061 

1.041 

1  .105 

1.  126 

I.04O 

2-5 

1.025 

1.038 

1.073 

1.043 

1.118 

1.  146 

I-°57 

3-o 

0.992 

1.044 

1.082 

1.044 

1.128 

1.163 

I  .072 

3-5 

0.966 

1.049 

i  .090 

1.045 

1.136 

I-I77 

1.085 

4-0 

0.944 

I-°53 

1.097 

1.046 

1.144 

1.190 

1.097 

NOTE  :  See  cuts  on  opposite  page. 


EMPIRIC  COEFFICIENTS 


99 


y//////M//////////^ 


////////////M^^^ 


100 


HYDRAULICS 


TRAPEZOIDAL    WEIRS. 


TABLE    X. 
VALUES  OF  p  FOR  TYPES  A  TO  E  FOR  SAME  b  AND 


H 

ft. 

Type  A. 

Type  B. 

Type  C. 

Type  D. 

D  with 
Vacuum. 

Type  E. 

E  with 
Vacuum. 

o-5 

0.968 

i  .060 

1.043 

1.069 

1.  088 

1.069 

1.069 

I.O 

1.071 

1.079 

1.040 

1.079 

1.106 

1.079 

1.079 

i-5 

1.077 

1.091 

1-037 

1.084 

1.117 

i.  088 

1.092 

2.0 

1.081 

1.096 

1.027 

I-OS7 

1.092 

1.063 

1.083 

2-5 

1.077 

1.093 

1.015 

1.041 

1.079 

1.049 

1.081 

3-o 

1.074 

1.090 

1.005 

1.028 

1.068 

1.039 

i.  080 

3-5 

1  .071 

1.087 

0.996 

1.018 

1.059 

i  .029 

1.079 

4.0 

1.069 

1.085 

0.989 

1.009 

1.051 

I.  021 

1.078 

EMPIRIC  COEFFICIENTS 


101 


COMPLEX  WEIRS. 

-38.-S4- 


TABLE    XI. 
VALUES  p  FOR  TYPES  M  TO  P. 


H 

ft. 

Type  M. 

Type  N. 

Type  0. 

Type  P. 

°-5 

0.964 

0.897 

1.095 

o  .920 

I  .0 

0.965 

0.946 

1.088 

0.915 

!-5 

0.963 

0.999 

1.084 

0.914 

2  .0 

0.949 

1  .025 

1.069 

°-935 

2-5 

0-933 

1.039 

1  .051 

0.950 

3-° 

0.920 

1.052 

i-°35 

0.962 

3-5 

0.911 

1.063 

i  .024 

0.972 

4.0 

0.903 

1  .072 

1.014 

0.982 

APPENDIX   A. 

A     COLLECTION      OF     WEIR    FORMULA     PROPOSED     BY 
DIFFERENT   AUTHORS,    WITH    DISCUSSION. 

i.    COMPLETE    OVERFALLS. 

THE  following,  Fig.  i,  shows  the  lettered  dimensions  used  in  the 
formulae  for  complete  overfalls. 


Fig.   i 


Eytelwein  in  his  "Handbuch  der  Mechanik  fester  Koerper  und 
der  Hydraulik,"  1823;  also, 

Weisbach  in  "Huelse's  Maschinen-Encyclopaedie,  "  1841,  give 
the  following  formula: 


2  g 


g 


Let  A2L  represent  the  level  of  a  quiet  reservoir  such  that  JGl  = 
2  g          °' 


102 


APPENDIX  A  103 

Then  according  to  Eq.  (i),  Chap.  I,  the  discharge  through  an 
opening  of  height  HQ  would  be 


3 

and  taking  off  the  small  quantity  passing  through  LE  and  which 
is 

_2    bV—/v*\* 

3  W' 

the  final  discharge  through  E^E  becomes 

Q  =  Ql  —  q  =  the  above  formula  (i). 

The  Eytelwein-Weisbach  formula  is  thus  derived  from  the  funda- 
mental Eq.  (i),  for  flow  through  a  lateral  orifice,  and,  therefore,  does 
not  apply  to  discharge  with  initial  velocity  of  approach. 

The  four  following  proofs  of  the  incorrectness  of  the  Eytelwein- 
Weisbach  formula  are  now  presented. 

1 .  No  consideration  is  given  to  width  and  depth  of  the  approach 
channel;  to  the  height  k  of  the  weir,  nor  to  the  shape  of  the  weir. 
All  of  these  circumstances  are  known  to  exert  a  very  considerable 
influence  on  the  discharge. 

2.  The  flow  of  approach  is  supposed  to  exert  an  hydrodynamic 
pressure  of  —  making  the  total  hydraulic  pressure  on  the  discharge 

section  (H  H ).     It  was  shown,  however,  that  this  depends  on 

V          2  gl 

H,  by  Bf  and  k,  and  may  have  any  value  between  4  ( — )   and 

\2  gl 
lS(fg 

3.  These  authors  were  undoubtedly  cognizant  of  some  of  these 
defects  in  their  formula  and  sought  to  correct  the  error  by  the 
coefficient  /*.     However,  a  rational  formula  must  express  the  true 
influence  of  all  the  hydrostatic  and  hydrodynamic  pressures  and 


104  HYDRAULICS 

weir  dimensions  on  the  discharge.  The  coefficient  n  should 
serve  merely  to  rectify  the  theoretically  correct  discharge  to  cover 
the  unknowable  effects  due  to  adhesion,  cohesion,  friction  and 
contraction. 

4.  The  values  of  ;*,  being  variable  in  character,  are  necessarily 
without  value  except  within  the  scope  of  the  experiments  from 
which  they  were  obtained.  For  rational  formulae  this  should  not 
be  the  case,  at  least  not  to  any  great  extent.  This  becomes  the 
more  important  when  it  is  realized  that  all  hydraulic  experiments 
must  be  confined  to  reasonably  small  conditions,  and  unless  a 
rational  solution  of  hydraulic  problems  is  made  possible  the  ulti- 
mate solution  applicable  to  extensive  waterpower  plants  would 
remain  impossible. 

The  above  Eytelwein-Weisbach  formula  was  discussed  at  some 
length  because  it  is  in  most  general  use  and  also  because  the 
objections  cited  will  be  found  equally  applicable  to  most  other 
formulae. 

Navier  proposed  the  following  formula,  wherein  the  head 
EO  =  0.2753  &  (see  Fig.  i),  a  relation  based  on  the  doctrine  of 
least  work.  The  formula  is 

Q  =  \lA  VTg  [i  -  (0.27S3)1]  H!  =  2.5261  ftbH*  .    .   (2) 

This  is  so  manifestly  incorrect  that  scarcely  any  comment  is 
necessary.  Professor  Ruehlmann  remarks  of  this  formula  that 
it  does  not  agree  any  better  with  experiments  than  do  the  Scheffler 
and  Braschmann  modifications  of  the  Weisbach  formula. 

Lesbros  experimented  with  very  small  overfalls  (b  =  &  inches; 
k  =  20  inches;  B  =  12  feet)  and  computed  the  values  ft  from 
the  fundamental  formula 


which  applies  only  for  discharge  through  a  lateral  orifice  when 
there  is  no  velocity  of  approach.  He  determined  2000  different 
values  for  /£,  all  for  the  constant  condition  b/B  =  0.054.  These 
coefficients  are  entirely  without  value  and  it  is  difficult  to  under- 
stand how  any  person  could  expend  the  mentality  required  for 


APPENDIX  A 


105 


these  computations  when  the  futility  of  the  undertaking  must  have 
been  apparent. 

Weisbach  later  continued  his  experiments,  commenced  in  1842, 
and  established  the  phenomena  of  incomplete  contraction.  On 
these  experiments,  made  with  8-inch  wide  openings  through  the 
thin  wall  of  a  1  4-inch  wide  flume,  he  based  the  two  following 
formulae: 

When  b  <  B 


and  when  b  =  B 

Q.  - 


+  0-3693    r 


(3b) 


The  coefficients  f  /*  are  those  given  in  Poucelet-Lesbros'  tables 
based  on  8-inch  wide  overfalls. 

Here  again  the  second  factor  represents  the  discharge  through 
a  lateral  orifice  and  the  first  factor  is  a  variable  coefficient  of  irra- 
tional form.  Besides  the  apparent  incorrectness  of  the  formula 
it  is  certain  that  coefficients  derived  from  8-inch  openings  are  not 
applicable  to  Weisbach'  s  experiments. 

Boileau,  in  1845,  was  induced  to  make  other  experiments  on 
complete  overfalls  for  cases  where  b  =  B  =  0.94  to  5.31  feet. 
He  gave  the  following  formulae  for  discharge  and  /*: 


or 


•     .     (4) 


where  e  =  OE,,  Fig.  i. 


106  HYDRAULICS 

It  is  clearly  seen  that  Boileau  did  not  consider  the  hydrodynamic 
pressure  against  the  discharge  area  and  the  weir,  nor  the  velocity 
of  approach.  Instead,  his  coefficient  is  made  to  cover  the  varia- 
tions due  to  these  pressures,  the  weir  dimensions  and  contractions, 
etc.  As  previously  shown,  such  formulae  cannot  have  any  general 
applicability. 

Redtenbacher ,  in  1848,  proposed  an  empiric  formula  which  was 
based  on  Castel's  experiments  on  complete  overfalls  0.4  to  29 
inches  wide.  He  gave 


(5) 


Q  =    0.381  +  0.062 bH  V2  gH 


and  when  b  =  B,  Q  =  0.443  bH  \/2  gH. 

According  to  Redtenbacher  these  formulae  are  applicable  only 
when  BT  =  5  bH,  and  b  is  at  least  equal  to  B/$  and  k  —  T1  equals 
at  least  2  H.  Finally  the  weir  must  be  sharp  crested. 

No  consideration  is  given  to  hydrodynamic  pressures  due  to  the 
flow  of  approach,  nor  to  the  dimensions  of  the  weir.  The  first 
term,  which  takes  the  place  of  /*,  is  a  constant  for  all  values  of 
b/B  =  constant.  It  is  hardly  necessary  to  point  out  the  uselessness 
of  this  formula. 

J.  B.  Francis,  in  discussing  his  epoch-making  "  Lowell  Hy- 
draulic Experiments,"  1855,  modified  Weisbach's  formula  to 
obtain  the  following: 

Q  =  3-33  0-  o-io  nH)H* (6) 

wherein  n  =  number  of  end  contractions; 

H  =  measured  height  of  water  above  the  weir  crest; 
H0  =  a  pressure  height  corrected  for  velocity  of  approach 

and  given  by  Francis  as 


In  these  experiments  B  =  13.96  feet;  k  =  2.04  feet  to  5.05  feet; 
and  b  =  9.995  to  9.997  feet.  The  weir  crest  was  the  sharp  plate 
since  adopted  as  the  standard  form  for  experimental  weirs. 


OF   THE 

UNIVERSITY    | 

OF 


APPENDIX  A  lO/ 


The  coefficient  3.33  in  Eq.  (6)  is  the  average  of  values  ranging 
between  3.3002  and  3.3617  and  is  a  value  for  f  /*  \/2  g.  Hence 
j«  =  0.6228  is  considered  constant  for  all  weir  dimensions  and 
depths  of  water,  which  is  certainly  wrong. 

When  there  are  no  end  contractions,  b  =  B  and  n  =  o,  and 
Eq.  (6)  becomes 

2  =  3-33^0*   .     .    .:.  .,.;..    .     (66) 

which  cannot  be  regarded  as  a  general  law  because  when  b  =  o.  10  nH 
in  Eq.  (6),  then  Q  =  o,  which  is  an  apparent  contradiction. 

These  objections,  together  with  the  assumption  that  the  flow 
takes  place  over  a  height  H0  while  in  reality  the  height  is  only  H, 
render  the  formula  quite  valueless  except  in  special  cases  resembling 
the  Francis  experiments. 

This  in  no  wise  vitiates  the  high  value  which  the  very  accurate 
experiments  of  Mr.  Francis  possess,  irrespective  of  any  theoretic 
deductions  which  may  now  or  have  ever  been  drawn  therefrom. 

Braschmann,  in  1861,  proposed  a  formula  based  on  the  principle 
of  least  work.  It  was  a  modified  form  of  Navier's  formula  using 
Castel's  and  Lesbros'  experiments  for  the  determination  of  his 
coefficients.  The  general  form  is 


Q  =  fibH  V2  gH  where  ft  =  0.3838  +  0.0368  -  +  t'^jj&>  .     (7) 

The  objections  to  this  formula  are  apparent  from  the  preceding. 
Bornemann,  in  1870,  experimented  with  weirs  for  which  b  = 

TT 

B  =  3.8  feet;  H  =  2.75  to  8.27  inches;  and    —  =  0.2  to  0.8. 

His  formula  is, 

for  #  <  -  T,    Q=  (0.5673  - 0.1239  y~) bH  vTp 


for 


H  >  -  r,    Q  =  (0.6402  -  0.2862  y  —^  b(H  +  z) 


(8) 


108  HYDRAULICS 

Bornemann  himself  points  out  that  his  formulae  are  not  appli- 
cable unless  b  =  B,  and  expresses  the  hope  that  somebody  may 
eventually  succeed  in  deriving  mathematically  correct  forms. 
The  first  formula  does  not  include  velocity  of  approach  and  the 
second  formula  does  this  by  introducing  the  fictitious  height  H  +  z. 

G.  Hagen  in  his  book  "Die  Stroeme,"  1871,  adopts  the  Eytel- 
wein-Weisbach  formula. 

M.  Becker,  in  1873,  gave  tne  following  formula: 

For  velocity  of  efflux: 


and  for  quantity 


....    .     .     (9) 


This  formula  is  based  on  the  incorrect  assumption  that  the 
velocity  of  approach  exerts  an  hydrodynamic  pressure  against 
the  discharge  area  only  and  that  the  mean  velocity  of  efflux  over  the 

total  depth  corresponds  to  a  pressure  head     —  H  +  —  ,  which 

9  2g 

cannot  be  generally  admitted. 

K.  Pestalozzi,  gives  the  Eytelwein-Weisbach  formula,  which  need 
not  be  repeated  here. 

Ruehlmann  expresses  the  opinion  that  the  scientific  value  of  all 
the  formulas  above  given  is  very  small.  He  believes,  however,  that 
they  are  applicable  to  cases  closely  resembling  the  experiments  on 
which  they  were  founded. 

After  presenting  numerous  examples  he  shows  that,  even  within 
range  of  the  experiments,  the  discharges  found  by  formulas  3,  4, 
6,  7  and  8  differ  by  amounts  varying  from  12  to  19  per  cent. 

Bazin.  The  objections  previously  cited  with  reference  to  the 
Weisbach  formula  apply  equally  to  the  following  Bazin  formula, 
1898,  Ann.  d.  Fonts  et.  Ch.,  p.  223,  where 


(10) 


APPENDIX  A 


109 


Cipolletti  uses  a  modification  of  Weisbach's  formula,  also  a 
simple  form  of  the  Francis  formula,  and,  in  a  very  unscientific 
manner,  adapts  these  to  some  experiments  of  his  own. 

The  attempt  to  use  these  experiments  for  the  determination  of 
fj.  in  the  foregoing  chapter  proved  futile  because  the  necessary 
weir  measurements  were  not  published  in  the  report  describing 
the  experiments. 

Fteley  and  Stearns,  as  a  result  of  their  very  extensive  hydraulic 
experiments,  made  in  1877  to  1879,  published  the  following  modi- 
fication of  the  Francis  formula: 

Q  =  3.31  m*  +  0.007  * (11) 

The  addition  of  the  last  term  alone  distinguishes  this  from 
Eq.  (66)  and"  hence  the  criticisms  previously  made  to  the  Francis 
formula  also  apply  here. 

2.    INCOMPLETE    OVERFALLS. 

Dubuat,  under  suppositions  discussed  in  Chapter  I,  derived  the 
following  formulae  and  gives  /*  =  fit  =  0.633: 

Q  =  f  f*bHt  V^lT2  +  fiJfH,  \/2~^T2  .     .     (12) 

A  CD 


Fig.  2 

Redtenbacher  says  that  the  derivation  of  formulae  for  incom- 
plete overfalls  is  connected  with  unsurmountable  difficulties  and 
adopts  Dubuat's  formula,  making  f  /i  =  0.57  and  /^  =  0.62 
both  constant. 


HO  HYDRAULICS 

The  first  half  of  Eq.  (12)  is  incorrect,  because  it  is  based  on  the 
assumption  that  discharge  through  the  height  H2  takes  place  as  for 
discharge  into  open  air,  which  is  not  true.  The  second  term  does 
not  include  velocity  of  approach  nor  suction  due  to  velocity  of 
discharge.  Ruehlmann  advises  against  the  use  of  these  formulae. 

Lesbros  based  the  following  simple  formula  on  his  experiments 
made  in  1829  to  1834. 


Q  =  -tibH     ^H2   ......     (13) 

where  /J.  is  variable  and  made  to  depend  on  the  ratio  H2/H.  In 
principle,  this  formula  is  entirely  wrong,  assuming  as  it  does  that  the 
discharge  takes  place  over  the  whole  height  H  and  with  a  uniform 
velocity  \/2  gH2. 

Bornemann,  as  a  result  of  his  experiments,  made  from  1866  to 
1872,  on  overfalls  22  to  45  inches  wide,  gives  this  formula: 


I'll  (H 

in  which   /*  =  0.702  —  0.2226  \  ~j-*  +  0.1845 

and          ^  =  ^(£)2. 

2g         2g\bTl 

The  first  member  of  Eq.  (14)  would  be  true  provided  the  upper 
discharge  did  take  place  through  the  height  H2  into  free  air.  The 
second  term*  is  incorrect  because  it  assumes  the  submerged  section 
as  discharging  into  quiet  water. 

G.  Hagen  gives  no  formula  for  incomplete  overfalls  but  states 
that  the  upper  layer  of  flow  may  be  regarded  as  a  complete  over- 
fall and  that  the  submerged  portion  is  subjected  to  a  uniform 
pressure,  corresponding  to  the  height  Hv  thus  leaving  out  of  con- 
sideration the  suction  and  assuming  the  counterpressure  from  the 
lower  level  active  over  the  entire  submerged  area. 

M.  Becker  proposes  the  formula 


(15) 


APPENDIX  A  III 

The  first  term  is  based  on  the  wrong  assumption  that  the  flow 
of  approach  exerts  an  hydrodynamic  pressure  on  the  discharge 
area  only  and  that  the  mean  discharge  velocity  corresponds  to  a 

pressure  height  ( —  H9  +  ).     The  second  term  neglects  suction 

\9  2  gl 

and  assumes  that  the  counterpressure  is  active  over  the  whole  sub- 
merged section. 

K.  Pestalozzi's  formula,  for  which  he  gives  /J.  =  0.8  to  0.85  and 
/*!  =  0.62,  is  patterned  after  the  Weisbach's  formula  for  complete 
overfalls.  It  is 


(16) 

The  second  term  neglects  suction  effect  and  weir  dimensions. 

Messrs.  Fteley  and  Stearns  contributed  a  new  formula  for 
incomplete  overfalls  which  is  purely  empiric  and  assumes  both 
the  upper  and  lower  pools  in  a  quiescent  state.  This  formula  is 


H 


wherein  c  is  a  coefficient  depending  on  the  ratio  H/Hl  and  varies 
between  3.089  and  3.372. 

While  it  is  not  supposed  that  any  empiric  formula  will  apply 
outside  of  the  limits  of  the  experiments  for  which  it  was  proposed, 
it  is  not  necessary,  in  view  of  what  has  gone  before,  to  say  more 
than  that  this  formula  has  served  its  purpose  well. 

It  was  not  until  the  experiments  of  Messrs.  Francis,  Fteley  and 
Stearns  had  been  made  and  given  to  the  world  that  anyone  had 
even  the  right  to  claim  any  great  knowledge  regarding  the  science 
of  weir  hydraulics.  While  there  is  still  far  more  to  do  along  the 
experimental  line  than  the  sum  total  work  already  accomplished, 
everyone  must  cherish  a  feeling  of  gratitude  towards  these  gentle- 
men and  express  the  fond  hope  that  some  day,  some  one  else  will 
continue  this  work. 


112  HYDRAULICS 

Several  large  hydraulic  laboratories  have  come  into  being  during 
recent  years  and  they  should  undoubtedly  contribute  something 
to  our  present  knowledge  which  will  furnish  such  data  as  are 
necessary  for  work  of  the  kind  here  treated. 

In  concluding  this  subject  it  may  be  stated,  that  without  excep- 
tion, modern  writers  have  adopted  one  or  more  of  the  formulae 
given  in  this  Appendix,  sometimes  without  mentioning  the  source, 
hence  this  review  is  considered  sufficiently  exhaustive  to  show, 
without  question,  the  irrational  constitution  of  all  older  formulae. 

The  attempt  here  made  to  offer  something  in  a  progressive 
direction  would,  therefore,  seem  justified.  However,  it  is  frankly 
admitted  that  the  subject  treated  is  still  a  very  imperfect  branch 
of  engineering  science. 


APPENDIX    B. 

ON  THE  FLOW  OVER  A  FLIGHT  OF^PANAMA  CANAL  LOCKS. 

Solution  of  a  Novel  Hydraulic  Problem. 

IN  the  design  of  a  movable  dam  for  the  head  of  the  triple  flight 
of  locks  at  Gatun,  one  of  the  first  and  most  perplexing  problems 
encountered  was  to  determine  the  conditions  of  flow  which  would 
obtain  in  the  event  of  serious  accident  to  any  of  the  upper  lock 
gates. 

Such  a  catastrophe,  while  highly  improbable,  is  nevertheless 
not  impossible,  and  in  the  remote  case  of  its  happening  would 
cripple  the  operation  of  the  entire  Panama  Canal  until  the  result- 
ing torrent  pouring  down  through  the  locks  could  be  effectively 
checked. 

To  meet  this  extraordinary  contingency,  it  is  proposed  to  erect 
a  movable  dam,  probably  of  the  swing  bridge  type.  Omitting 
the  essential  details  of  the  dam,  it  is  sufficient  to  mention  here  that 
the  strength  of  the  structure  and  the  power  for  its  operation  depend 
for  their  determination  on  the  depth  and  velocity  of  flow  through 
the  section  at  which  the  dam  is  to  be  located. 

For  some  time  this  flow  problem  seemed  to  offer  unsurmountable 
difficulties  owing  to  the  many  unknown  quantities  which  neces- 
sarily enter.  A  thorough  search  through  the  world's  hydraulic 
literature  did  not  add  much  encouragement,  and  it  became  manifest 
that  a  solution,  if  one  were  possible,  would  mean  a  radical  depar- 
ture from  any  of  the  previously  known  methods  for  solving 
hydraulic  problems.  Also  a  solution,  to  be  of  any  value  in  con- 
nection with  the  general  question,  must  lay  claim  to  considerable 
accuracy. 

The  triple  flight  of  locks  at  Gatun  presents  a  succession  of 


1  14  HYDRAULICS 

weirs  or  overfalls  which  may  be  complete  or  incomplete,  depend- 
ing on  the  profile  of  the  locks  and  the  total  drop  in  the  water 
levels  between  Gatun  Lake  and  the  Atlantic  Ocean. 

For  the  discharge  through  the  straight  portions  of  the  locks  and 
canal  the  Chezy  formula  with  Bazin's  coefficients  will  be  used. 
This  is  probably  the  most  reliable  for  flow  through  open  channels 
having  steep  slope.  For  small  slopes,  the  well  tested  formula  of 
Ganguillet  and  Kutter  may  be  more  accurate. 

Suppose  now  that  acceptable  formulae  are  at  hand  for  finding 
discharge  over  any  single  drop,  also  for  the  straight  portions  of  the 
canal  and  locks.  This  then  furnishes  one  formula  for  each 
condition  of  flow  throughout  the  entire  stretch  of  canal  under 
discussion.  Hence  there  are  as  many  possible  equations  as  there 
are  varieties  of  conditions  of  flow. 

Regarding  the  feasibility  of  solving  any  or  all  of  these  equations, 
it  will  be  best  to  give  the  general  forms  and  discuss  them  with 
relation  to  the  known  and  unknown  quantities. 

For  the  straight  portions  of  the  channel  and  locks  the  Chezy 
formula  gives 

v  =  CVrs  and   Q  =  Av  =  AC  V7s     .     .     .     (i) 
also  sL  =  H,  which,  substituted  into  Eq.  (i),  gives 


Q  =  4C-M       .  wherein  C  =  -    -    ....     (2) 

0.552  +  ™ 

Q  =  quantity  of  discharge  in  cubic  feet  per  second. 

A  =  discharge  section  in  square  feet. 
r  =  mean  hydraulic  radius  in  feet. 

L  =  length  of  straight  channel  in  feet. 

H  =  fall  in  surface,  in  feet,  over  length  L. 

C  =  the  Bazin  Coefficient,  wherein  m  is  an  experience 
number  the  values  of  which  are  given  by  Bazin  for  all  conditions 
of  flow.  The  values  of  m  vary  from  0.06  to  1.75,  see  under  2, 
Chapter  VI. 


APPENDIX  B  US 

For  incomplete  overfalls,  the  new  formulae,  Eqs.  (28),  give 

(3) 


ij1  nV2 

wherein  S  =  —  }  S.  =  S  +  H2  H  -- 

2  g  2  g 


Here  b  =  uniform  width  of  canal  and  overfall  in  feet. 

g  —  acceleration  of  gravity  =  32.16  feet. 

v  =  mean  velocity  of  approach  in  feet  per  second. 

V  =  mean  velocity  of  discharge  in  feet  per  second. 

n  =  coefficient  of  contraction  =  0.67. 

k  =  height  of  weir  above  approach  canal  bottom. 
Hl  =  depth  of  weir  crest  below  lower  pool. 
H2  =  depth  of  lower  pool  below  upper  pool. 

T  —  depth  of  approach  canal. 
7\  =  depth  of  discharge  section. 

/JL  and  /^  are  discharge  coefficients    for  free    and    submerged 
discharge  respectively. 


Then  v=       =      .;    F  =      -;andJffI  =  r-#2.    .    .     (4) 

The  known  quantities  are  b  =  100  feet;  T  =  50  feet;  and  k, 
being  small  compared  with  the  depth  T,  is  neglected.  All  other 
quantities  are  unknown  and  depend  for  their  values  on  Q  and 
H2.  Hence,  if  Q  and  H2  are  regarded  as  the  independent  vari- 
ables, the  other  quantities  may  be  expressed  in  terms  of  these  two. 
See  Plate  I,  left  hand  end  of  lower  profile,  for  lettered  dimensions. 

The  upper  profile  of  Plate  I  represents  the  condition  prior  to  an 
accident,  the  lower  profile  gives  the  computed  water  surface  down 
the  flight  of  locks  after  a  uniform  condition  of  flow  has  been 
established. 


1 1 6  HYDRAULICS 

While  it  is  possible  then"  to  substitute  numerical  values  into 
equations  (3),  involving  only  Q  and  H2  as  the  final  unknowns, 
it  is  quite  impossible  to  solve  directly  the  complicated  form  which 
results  from  such  substitutions. 

The  approach  velocity  and  quantity  for  the  second  drop  is  now 
represented  by  the  discharge  velocity  and  quantity  from  the  first 
drop.  Hence  if  the  first  case  were  solved  the  second  could  be 
solved  in  like  manner,  and  so  on  for  a  third  or  fourth  drop. 

Now  the  quantity  of  discharge  for  a  continuous  flow  must  be 
constant,  hence  there  is  only  one  finally  unknown,  Q,  while  there 
is  an  unknown  H2  for  each  drop.  This  then  enables  writing  out 
one  equation  for  each  H2  in  terms  of  Q,  in  which  Q  is  a  function 
of  H2  and  itself.  Thus: 

Q  =  f(Q,  H2)  for  first  drop; 

Q  =  f(Q,  H2')  for  second  drop;      .     .V    .     (5) 

Q  =  T  (Q,  H2")  for  third  drop; 

wherein  there  is  one  more  unknown  quantity  than  the  number  of 
equations.  Hence  the  problem  is  not  solvable  until  one  other 
equation,  involving  these  unknowns,  is  given. 

In  the  same  manner  a  series  of  equations  may  be  written  out 
for  the  horizontal  stretches  of  the  canal  by  using  Eq.  (2),  thus: 


IY  Tlf  rr  13"  I  r"  ~H'" 

=Ac  \A--  =  A'c'  V^-    =  A'V  V(££L  ,  etc.,  (6) 


wherein  all  the  quantities  are  known  except  Q,  H',  H"  and  H'", 
and  these  equations  also  number  one  less  than  the  number  of 
unknowns. 

However,  the  ocean  level  and  the  level  for  Gatun  Lake  being 
fixed,  relatively,  the  total  difference  in  their  levels  being  87  feet, 
the  final  condition  follows: 

1H2  +  *H'  =  87  .......     (7) 

Hence  putting  Eqs.  (5),  (6)  and  (7)  together  this  will  give  as 
many  equations  as  there  are  unknowns,  so  the  problem  is  definitely 


APPENDIX  B  117 

solvable.  But  owing  to  the  complexity  of  the  equations  there  is 
no  method  known  in  algebra  by  which  these  equations  can  be 
solved  for  simultaneous  values  of  the  unknowns. 

The  solution  given  in  the  following  is  believed  to  be  new  and  is 
original,  as  nothing  bearing  on  this  point  could  be  found  in  any 
literature  extant. 

After  much  deliberation  and  study  it  was  found  that  simulta- 
neous values  for  the  unknowns  were  obtainable  by  a  graphic 
representation  of  Eq.  (7),  inasmuch  as  it  is  a  straight  line  equa- 
tion and  depends  for  its  fulfilment  on  a  certain  definite  value 
of  Q.  Then  if  the  values  H2  and  Hf  are  ascertained  for  all 
reasonable,  assumed  values  of  Q,  and  plotted  as  co-ordi- 
nates, there  will  result  as  many  curves  as  there  are  equations 
less  one. 

The  missing  equation  is  Eq.  (7)  and,  by  trial,  such  a  vakie  of 
Q  can  be  found  for  which  Eq.  (7)  will  be  satisfied,  and  this  fur- 
nishes the  final  solution. 

The  value  of  Q  thus  determined,  all  the  particular  values  of 
H2  and  H'  become  known  and  the  profile  of  the  surface  can  be 
drawn.  This  also  fixes  the  velocity  for  every  section  along  the 
entire  stretch  of  canal. 

To  exemplify  the  above  reasoning,  the  complete  solution  will 
now  be  illustrated  by  reference  to  Plates  I  and  II. 

Since  all  of  the  equations  (3)  are  so  extremely  involved  and 
complicated  that  they  are  not  directly  solvable,  it  becomes  neces- 
sary to  assign  values  to  Q  and  find  values  for  H2  by  .successive 
trials,  continuing  this  process  until  the  equation  is  satisfied.  While 
this  is  a  laborious  operation  it  is  far  easier  than  at  first  appears 
and  with  a  little  experience  an  average  computer  can  soon  learn 
to  solve  a  point  by  two  or  three  approximations. 

A  rough  idea  as  to  the  limits  between  which  the  unknown  Q 
may  be  located,  can  be  obtained  by  a  preliminary  inspection  of 
the  given  conditions,  and  by  observing  that  a  maximum  value  for 
V  =  0.67  \/2  gh.  However,  this  may  be  twice  as  large  as  the 
real  velocity. 


Il8  HYDRAULICS 

In  the  present  problem  it  was  considered  safe  to  assume  that 
the  first  velocity  of  approach  would  have  a  value  somewhere 
between  20  and  25  feet  per  second,  although  the  theoretic  v 
would  be  about  37  feet. 

It  was  also  reasonable  to  suppose  that  each  overfall  would  be 
incomplete,  as  a  careful  inspection  of  the  profile  would  lead  one 
to  foresee.  Hence,  the  formula  for  incomplete  overfalls  was 
used. 

But  this  assumption  might  have  been  erroneous,  a  fact  which 
would  be  clearly  indicated  by  the  values  H2  resulting  from  a  few 
preliminary  computations.  In  the  latter  event  the  formula  for 
complete  overfalls  would  have  to  be  employed. 

Suppose  then  that  we  have  chosen  the  appropriate  Eq.  (3)  and 
that  the  required  Q  corresponds  to  some  velocity  between  20  and 
25  feet.  Also,  assume  values  for  v  from  20  feet  and  up,  one  foot 
apart;  this  will  give  a  sufficient  number  of  points  to  plot  a  curve 
such  as  shown  on  Plate  II,  for  the  upper  lock,  drop  i. 

From  Eqs.  (4),  Q  may  be  found  for  any  assumed  v  when  the 
section  is  known,  and  since  the  depth  at  the  head  of  the  canal 
must  remain  constant,  the  discharge  section  may  be  assumed 
constant  at  the  entrance  to  the  canal.  Therefore,  the  assumed 
data,  for  which  values  of  H2  are  sought,  would  be  as  follows : 

Case  (i)  v  =  20  ft.  T  =  50  ft.  b  =    100  ft.  Q  =  100,000  cu.  ft. 

Case  (2)  21  50  loo  105,000 

Case  (3)  22  50  100  110,000 

Case  (4)  23  50  100  110,000 

Case  (5)  24  50  100  115,000 

Case  (6)  25  50  100  120,000 

The  complete  computation  for  the  first  point  will  now  be  given, 
and  this  will  serve  as  an  illustration  for  all  of  the  computations. 
The  problem  is  to  solve  Eq.  (3)  for  the  case  when  Q  =  100,000 
cubic  feet  per  second,  as  per  case  (i).  (See  Plate  I.) 

The  coefficient  /^  and  ft  must  first  be  found  from  Eq.  (7  7 a) 
by  substituting  proper  values  for  H2  and  b. 


APPENDIX  B  IIQ 

H2  is  not  known  but  since  the  total  of  the  three  drops  is  87 
feet,  it  is  sufficiently  close  to  assume  H2  =  25  feet,  as  the  term  in 
Eq.  (770)  involving  H2  has  very  small  influence  on  jj.  and  does 
not  enter  into  /^  Then  with  b  =  100  feet,  which  is  the  constant 
width  of  the  locks,  Eqs.  (770)  give 

.  -, 
+  0.000146  b  —  0.4150  or  fi  =  0.5533, 

Pi  =  0-5346  +  0.000146  b  =  0.5492. 

From  Table  VIII  it  is  seen  that  p  would  be  less  than  unity,  but 
as  no  experiments  on  submerged  weirs  were  available  and  as  the 
crest  in  our  problem  is  really  a  somewhat  narrow  sill  of  the  miter 
gates,  it  is  on  the  safe  side  to  assume  p  =  i. 

Hence  in  the  following  computations  these  coefficients  are  used  : 
n  =  0.67,  p  =  i  and  /*  =  /^  =  0.55. 

(1)    Assume  now  that  H2  =15  feet  when  Q  =  100,000  cubic 

feet.    Then  v  =     IOO?OO°   =  20  feet;  -H.  =  T  -H2  =  ^  feet;  and 
50  X  100 

T   =  H.  +  21  =  56  feet,  from  which   V  =  ^-  =  17.86;  —  = 

bT,  2  g 

3.32    and    $\/*g  =  441.375.     Also    S  =  -  -   =  6.20  and  5t  = 

nV2 
S  +  H2  +  -    -  =  6.20  +  15  +  3.32  =  24.52.     When  k  =  o,  then 

5j  =  S2  and  y  —  l  --  -2  =  \//Sl  ,  hence  Eq.  (3)  becomes 


-V-     .    (8) 

Substituting  all  the  above  values  into  Eq.  (8)  and  solving,  then 
Q  =  441-375  [f  (121.4  -  15-4)  +  (35  -3-32)  ^24.52  ]=  100,420  .  (9) 

(2)    This  indicates  that  the  first  assumption  for  H2  was  a  little 
large  and  the  operation  is  repeated  for  H2  =  14.75  feet-     It  should 


120  HYDRAULICS 

be  noted  that  this  change  does  not  affect  S  and  that  a  second  com- 
putation is  much  easier.     The  new  value  gives 


5,  =  5  +  H2  +          =  24.24;  H,  =  35.25;    7\  =  56.25;  and 

V  =  17.78.     Hence 

Q  =  44i-375[f  (H9-4  -i5-4)  +  (35-25  -3-288)  ^24.24]  =  99989. (10) 

The  exact  value  of  H2  may  now  be  found  by  interpolation 
between  the  values  (9)  and  (10)  by  correcting  the  last  value  for 
ii  cubic  feet,  which  gives  H2  =  14.75,  because  the  correction 
would  be  in  the  fourth  decimal.  It  should  be  mentioned  here 
that  interpolation  is  not  permissible  unless  reasonably  close  values 
have  been  found  on  both  sides  of  the  true  value. 

In  this  manner  the  values  H2  in  Table  I,  for  drop  i,  were  found, 
and  these  when  plotted  gave  the  curve  for  discharge  for  upper  lock 
at  drop  i,  see  Plate  II. 

Now  each  one  of  the  above  assumptions  for  v  and  Q  (which 
always  fixes  a  definite  value  F,  for  velocity  of  discharge)  fur- 
nishes the  conditions  for  approach  to  the  second  drop,  drop  2. 
This  is  very  important  and  on  this  fact  is  based  the  simultaneous 
relation  of  Q  with  the  several  values  H2  subsequently  found  for 
each  drop. 

However,  the  discharge  in  passing  over  the  length  of  the  upper 
lock  must  have  some  slope  sufficient  to  continue  the  discharge 
from  the  first  drop.  This  slope  and  fall  Hf  over  a  distance 
L  —  1000  feet  is  now  computed  by  Eq.  (2)  for  each  v  above 
assumed,  and  these  figures  are  given  in  Table  I  in  the  horizontal 
column  Upper  Lock  (Hf)  and  plotted  as  curve  AB,  Plate  II. 

The  various  depths  Tv  being  the  depths  of  discharge  from  the 
•first  drop,  are  now  reduced  by  amounts  Hf,  giving  new  values 
Tf  =  Tl  —  H'y  from  which  the  new  velocity  of  approach  for  the 
second  drop  is  found  for  each  of  the  previously  assumed  values  of 
Q.  Hence,  by  using  the  values  T'  in  place  of  the  former  value 


APPENDIX  B  121 

T,  each  case  of  Q  may  again  be  solved  exactly  as  for  the  first 
drop.     From  Plate  I  the  following  values  may  be  taken: 

_2_ 


vf  =  — ^— ,;  Hi  =  T  -  H3';  Tj  =  Hi  +  31  and  V  = 
100  T 

Whence  the  same  computations  are  repeated  for  the  second  drop 
and  results  entered  in  Table  I  and  plotted  on  Plate  II,  as  the  dis- 
charge curve  for  drop  2. 

By  a  repetition  of  this  process  to  the  third  drop  the  new  values 
become 


and  after  computing  H"  for  each  assumed  Q,  the  discharge  curve 
for  drop  3  was  plotted  on  Plate  II. 

Finally,  from  Eqs.  (2)  or  (6),  the  falls  Hf  through  the  lower 
lock,  the  Approach  Canal,  and  two  miles  of  wide  canal  connect- 
ing with  the  ocean,  may  be  found  for  each  of  the  first  assumed 
quantities  of  discharge,  and  a  discharge  curve  may  then  be  plotted 
for  each  channel. 

Having  thus  found  the  related  discharge  curves  for  each  con- 
dition of  flow,  over  the  entire  stretch  of  canal  and  locks,  the  final 
solution  is  easily  accomplished.  Since  for  any  value  of  Q,  the 
accompanying  values  H2  and  H'  are  simultaneous  values,  made 
so  by  the  previous  method  of  computation,  then  such  a  value  can 
be  found,  by  trial,  which  will  make  ^H2  +  2H'  =  87  feet  and 
the  problem  is  solved. 

Referring  to  Plate  II,  and  the  tabulation  in  Table  I,  the  value 
Q  =  115,570  cubic  feet  per  second  satisfies  this  final  condition. 

The  figures  in  the  last  column  of  Table  I  were  read  from  the 
curves  of  Plate  II,  excepting  a  few  of  the  small  drops  which  could 
better  be  interpolated  from  the  table.  These  several  values  of 
H2  and  H'  were  then  used  to  plot  the  surface  slope  for  the  entire 
stretch  of  canal,  as  shown  in  the  lower  profile  of  Plate  I.  From 
this  profile  the  depth  and  velocity  of  flow  at  any  discharge  section 


122  HYDRAULICS 

may  be  found  by  dividing  the  area  of  the  section  into  115,570. 
The  upper  approach  velocity  is  thus  found  to  be  23.2  feet. 

Regarding  the  accuracy  of  the  above  method  it  is  believed  one- 
half  per  cent  would  cover  all  of  the  slight  inaccuracies  inherent  in 
the  solution  of  the  equations,  while  the  coefficients  are  to  some 
extent  speculative.  It  may  be  said  then  that  the  problem  is 
solved  with  the  greatest  accuracy  attainable  with  our  present 
knowledge  of  the  empiric  coefficients,  and  these  may  be  consider- 
ably in  error  when  the  differences  in  circumstances,  for  which  they 
were  determined,  are  considered. 

Hence,  we  may  conclude  that  such  problems  as  the  above  are 
susceptible  to  general  solution  within  the  knowable  accuracy  of 
the  empiric  coefficients.  This  then  emphasizes  the  importance 
of  conducting  larger  hydraulic  experiments  especially  for  cases 
of  deep  and  submerged  flow. 


APPENDIX  B 


123 


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HYDRAULICS 


APPENDIX  B 


125 


120,000 


115,570 
115,000 


ilO,000 


105,000 


100,000; 


DISCHARGE  CURVES  FOR  GATUN  LOCK  FLIGHT. PANAMA  CANAL, 
PLATE  II 


APPENDIX   C. 

A  TABULATION    OF   THE   NEW   FORMULAE. 
ARRANGED   FOR  READY   REFERENCE. 

NOTE  —  For  definitions  of  terms  see  p.  IX. 


Jji 

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a.  a. 


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11 


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mulae  for  Coefficients  M 
sed  with  Reserve.  All  for 
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ext  for 
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INDEX 

PAGE 

Backwater  conditions 46 

height  and  distance 46 

table  of  functions 49 

Bazin's   formula 58,   108 

Becker's   formula 108,  no 

Boileau's   formula 105 

Bornemann's  formula 107,  no 

Braschmann's  formula 107 

Chezy's  formula 58 

Cipolletti's  formula 109 

Coefficients  for  weir  formulae 75  to  101 

Complete  overfall  weirs 2,   n,  20,   77,  102 

Contraction 2,  76 

complete 2,  76 

end 2,  95 

partial 2,  76 

Cornell  University  experiments 81,  82 

Curved  weirs 23 

Definitions  of  terms ix 

Derivation  of  new  formulae  for  diversions 52,  61,  67,  70 

incomplete  overfalls 28 

normal,  complete  overfalls 1 1 

oblique,  complete  overfalls 20 

sluice  weirs 36 

Dubuat's  formula 109 

Empiric  coefficients 75  to  101 

End  contraction 2>  95 

Examples  for  backwater  problems 50 

133 


134  INDEX 

PAGE 
Experiments  by  Boileau 80,  90 

Bornemann 90 

Engerth   90 

Francis 79,  80,  83,  86 

Fteley  and  Stearns 79,  83 

Lesbros 80,  90 

Rafter 81 

Weisbach 90 

Williams 95  to  101 

Cornell  University 81,  82 

Eytelwein's  formula 102 

Flow  in  rivers  and  canals 52 

over  flight  of  Panama  Canal  locks 113  to  125 

through  lateral  orifices 4 

Formulae  for  backwater  height  and  distance 47 

complete  overfalls 18,  19,  128 

curved  complete    overfalls 24,  129 

incomplete  overfalls 31,  32,  34,  130 

empiric  coefficients 77  to  101 

oblique,    complete   overfalls 23 

sluice  weirs  and  gates 40,  41,  43,  45,  132 

various   forms  of  complete  overfalls 23,  129 

waterpower  diversions 64,  65,  69,  70,  73,  129,  131 

Francis'   formula 106 

Fteley  and  Steam's  formula 109,  1 1 1 

Fundamental    equations 4,  6 

Ganguillet  and  Kutter's  formula 59 

Hagen's  formula  108,  no 

Hydraulic  pressure  defined i 

Hydrodynamic  pressure  defined i 

Hydrostatic  pressure  defined i 

Incomplete  overfall  weirs v 2,  28,  84,  109 

Lesbros'  formula 104,  no 

Modified  forms  of  complete  overfalls 23,  24,  26,  129 


INDEX  135 

PAGE 
Navier's  formula 104 

New  formulae  for  diversions 52,  61,  67,  70 

incomplete   overfalls 28 

normal,    complete   overfalls u 

oblique,  complete  overfalls 20 

sluice  weirs 36 

Overfall  defined 2 

Oblique  weirs 20 

Panama  Canal  locks 113  to  125 

Partial  contraction 2,  76 

Pestalozzi's  formula 108,  1 1 1 

Pier  obstructions " 33,  34 

Pressure,  hydraulic i 

hydrodynamic i 

hydrostatic i 

Proposed  solution  for  waterpower  diversions    57 

Redtenbacher's  formula 106,  109 

Ruehlmann's  formula 108 

Sluice  weirs  and  gates 2,  36,  89 

Solution  of  waterpower  problems 57 

Table  for  backwater  functions 49 

Terms  defined ix 

Velocity i 

Velocity  of  approach i 

Velocity  of  discharge 2 

Waterpower  canals 52,  57 

Weir  crests 95  to  i°i 

Weir  formulae,  coefficients  for 75  to  101 

Weirs,  curved 23 

oblique 20 

Weisbach's  formula . .  102,  105 


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*  Laboratory  Manual  for  Students I2mo,    i  oo 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) Svo,    2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) Svo,    2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) 12 mo,    i  oo 

4 


Holley  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments ,  and  Varnishes. 
(In  Press) 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  oo 

Iddings's  Rock  Minerals 8vo,   5  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  I  25 
Johannsen's  Key  for  the  Determination  of  Rock-forming  Minerals  in  Thin  Sec- 
tions.    (In  Press) 

Keep's  Cast  Iron 8vo,  2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis 12010,  i  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

*  Langworthy   and  Austen.        The   Occurrence   of  Aluminium  in  Vegetable 

Products,  Animal  Products,  and  Natural  Waters 8vo,  2  oo 

Lassar-Cohn's  Application  of  Some  General  Reactions  to  Investigations  in 

Organic  Chemistry.  (Tingle.) lamo,  i  po 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Lob's  Electrochemistry  of  Organic  Compounds.  (Lorenz.) 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments 8vo,  3  oo 

Low's  Technical  Method  of  Ore  Analysis 8vo,  3  oo 

Lunge's  Techno-chemical  Analysis.  (Cohn.) I2mo  I  oo 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i  50 

Maire ' s  Modern  Pigments  and  their  Vehicles .     (In  Press. ) 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2mo,  i  50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  i2mo,  60 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) I2mo,  i  25 

Matthew's  The  Textile  Fibres.    2d  Edition,  Rewritten    8vo,  400 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .i2mo,  oo 

Miller's  Manual  of  Assaying i2mo,  oo 

Cyanide  Process i2mo,  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) .  .  .  .  i2mo,  50 

Mixter's  Elementary  Text-book  of  Chemistry I2mo,  50 

Morgan's  An  Outline  of  the  Theory  of  Solutions  and  its  Results i2mo,  oo 

Elements  of  Physical  Chemistry I2mo,  3  oo 

*  Physical  Chemistry  for  Electrical  Engineers i2mo,  5  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Muir's  History  of  Chemical  Theories  and  Laws 8vo,  4  oo 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,  5  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) I2mo,  i  50 

"              "           "             Part  Two.     (Turnbull.) i2mo,  200 

*  Palmer's  Practical  Test  Book  of  Chemistry -. 12mo,  1  oo 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.     (Fischer. ) .  .  .  .  12010,  i  25 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) I2mo,  I  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis I2mo,  i  25 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Standpoint. . 8v o ,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  4  oo 

Disinfection  and  the  Preservation  of  Food. 8vo,  4  oo 

Riggs's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

5 


Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo> 

*  Whys  in  Pharmacy lamo,  i  oo> 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo§  3  00= 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) 8vo,  2  50- 

Schimpf's  Text-book  of  Volumetric  Analysis i2mo,  2  50 

Essentials  of  Volumetric  Analysis.  .  .  „ ,  i2mo,  i  25 

*  Qualitative  Chemical  Analysis 8vo,  i    25 

Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students ' .  .  .  8vo,  2  50- 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco  3  oo^ 

Handbook  for  Cane  Sugar  Manufacturers i6mo.  morocco.  3  oo> 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

*  Descriptive  General  Chemistry 8vo»  3  oo- 

Treadwell's  Qualitative  Analysis.     (Hall.) • 8vo,  3  oo 

Quantitative  Analysis.     (Hall.) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 3vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

*  Walke's  Lectures  on  Explosives 8vo;  4  oo 

Ware's  Beet-sugar  Manufacture  and  Refining.     Vol.  I Small  8vo,  4  oo 

"  "  "  "          "  Vol.11 bmallSvo,  500- 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oa 

Weaver's  Military  Explosives 8vo,  3  oo 

Wehrenfennig's  Analysis  and  Softening  of  Boiler  Feed-Water 8vo,  4.  oo 

Wells's  Laboratory  Guide  in-Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students I2mo,  i  50 

Text-book  of  Chemical  Arithmetic I2mo,  i  25 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes I2mo,  i  50 

Chlorination  Process I2mo,  i  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry -  •  •  •  • i2mo,  2  oo 


CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS  OF    ENGINEERING 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments I2mo,  3  oa 

Bixby's  Graphical  Computing  Table Paper  io,£  •  24^  inches.  25 

Breed  and  Hosmer's  Principles  and  Practice  of  Surveying 8vo,  3  oo 

*  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal    .  .          8vo,  3  50 
Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

*  Corthell's  Allowable  Pressures  on  Deep  Foundations I2mo,  i  25 

Crandall's  Text-book  on  Geodesy  and  Least  bquares :  .0*0,  3  oo 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage i2mo,  i   50 

Practical  Farm  Drainage I2mo,  i  oo 

*Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  oo 

Flemer's  Phototopographic  Methods  and  Instruments 8vo,  5  oo 

Folwell's  Sewerage.      (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  Ives's  Stereotomy 8vo  2  50 

Goodhue's  Municipal  Improvements I2mo,  i  50 

Gore's  Elements  of  Geodesy 8vo»  2  5° 

*  Hauch  and  Rice's  Tables  of  Quantities  for  Preliminary  Estimates I2mo,  i  25 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 


Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,    2  50 

Howe's  Retaining  Walls  for  Earth 12010,    i  25 

Hoyt  and  Graver's  River  Discharge 8vo,     2  oo 

*  Ives's  Adjustments  of  the  Engineer's  Transit  and  Level i6mo,  Bds.         25 

Ives  and  Hilts's  Problems  in  Surveying i6mo,  morocco,     i  50 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,    4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,    2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.)  •  12010,    2  oo 
Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,    5  oo 

*  Descriptive  Geometry.    8vo,     i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,    2  50 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,    2  oo 

Nugent's  Plane  Surveying 8vo,    3  50 

Ogden's  Sewer  Design i2mo,    2  oo 

Parsons's  Disposal  of  Municipal  Refuse. 8vo,     2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,    7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,    5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,    4  oo 

Riemer's  Shaft-sinking  under  Difficult  Conditions.     (Corning  and  Peele.).  .8vo,     3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,     i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,    2  50 

•Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,    2  oo 

"Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,    5  oo 

Tracy's  Plane  Surveying I6mo,  morocco,     3  oo 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,    5  oo 

Venable's  Garbage  Crematories  in  America .8vo,     2  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  oo 

Sheep,    6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,    5  oo 

Sheep,    5  50 

Law  of  Contracts 8vo,    3  oo 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,    2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,     i  25 
Wilson's  Topographic  Surveying 8vo,    3  50 

BRIDGES  AND  ROOFS. 

Boiler's  Practkal  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations 8vo,  3  oo 

Design  and  Construction  of  Metallic  Bridges ".  .  .  .8vo,  5  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  oo 

Poster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

•Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Grimm's  Secondary  Stresses  in  Bridge  Trusses.     (In  Press.) 

Howe's  Treatise  on  Arches 8vo,  4  oo 

Design  of  Simple  Roof -trusses  in  Wood  and  Steel 8vo,  2  oo 

Symmetrical  Masonry  Arches 8vo,  2  50 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges : 

Part  I.     Stresses  in  Simple  Trusses , 8vo,  2  50 

Part  II.    Graphic  Statics >. ,* .  .  .8vo,  2  50 

Part  III.  Bridge  Design i-. '.'. 8vo,  2  50 

Part  IV.  Higher  Structures 8vo,  2  50 

7 


Morison's  Memphis  Bridge.  , 4to,  10  o» 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers .  .  i6mo,  morocco,  2  oo 

Specifications  for  Steel  Bridges i2mo,  50 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume 8vo,  3  50 

HYDRAULICS. 

Barnes's  Ice  Formation 8vo,  3  oo> 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.  (Trautwine.) 8vo,  2  oo- 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo- 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  so- 
Hydraulic  Motors.  . 8vo,  2  oo 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12010,  3  oo 

Folwell's  Water-supply  Engineering • 8vo,  4  oo 

FrizelPs  Water-power 8vo,  5  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo.  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.  (Hering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Clean  Water  and  How  to  Get  It Large  I2mo,  l  5o 

Filtration  of  Public  Water-supply .8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water- works 8vo,  2  50- 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

*  Hubbard  and  Kiersted's  Water- works  Management  and  Maintenance.     8vo,  4  co- 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  oo- 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo- 

Schuyler's   Reservoirs  for   Irrigation,   Water-power,   and   Domestic   Water- 
supply Large  8vo,  5  oo> 

*  Thomas  and  Watt's  Improvement  of  Rivers 4*0,  6  oo- 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams.     5th  Edition,  enlarged.  .  .4to,  6  oo> 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4to,  10  oo> 

Whipple's  Value  of  Pure  Water Large  i2mo,  i  oo* 

Williams  and  Hazen's  Hydraulic  Tables 8vo,  i  50 

Wilson's  Irrigation  Engineering Small  8vo,  4  oo- 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50* 

Elements  of  Analytical  Mechanics 8vo,  3  oo- 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo,  5  oo> 

Roads  and  Pavements 8vo,  5  oa 

Black's  United  States  Public  Works Oblong  4to,  5  oo- 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50- 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  5* 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo> 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  410.  7  5<> 

*Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 

8 


Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Graves's  Forest  Mensuration 8vo,  4  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Keep's  Cast  Iron. 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Strength  of  Materials i2mo,  i  0.0 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo»  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  oo 

*  Ries's  Clays:  Their  Occurrence,  Properties,  and  Uses 8vo,  5  oo 

Rockwell's  Roads  and  Pavements  in  France i2mo,  r  25 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vc,  3  oo 

*Schwarz's  Longleaf  Pine  in  Virgin  Forest  ...   i2mo,  i    25 

Smith's  Materials  of  Machines i2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement. i2mo,  2  oo 

Text-book  on  Roads  and  Pavements I2mo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Materials  of  Engineering.     3  Parts.. 8vo,  8  oo 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Turneaure  and  Maurer's  Principles  of  Reinforced  Concrete  Construction.   .8vo,  3  oo 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.).  .  i6mo,  mor.,  2  oo 

*  Specifications  for  Steel  Bridges.  .  . ,  T. I2mo,  50 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V. )  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and 

Steel.  ...                    8vo,  4  oo 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Crockett's  Methods  for  Earthwork  Computations.     (In  Press) 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco  5  oo 

Dredge's  History  of  the  Pennsylvania  Railroad:   (1879) Paper,  5  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .i6mo,  mor.,  2  50 
Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Raymond's  Elements  of  Railroad  Engineering.     (In  Press.) 

9 


Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral. i6mo,  morocco,  z  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  x  so 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo 

Economics  of  Raikoad  Construction Large  i2mo,  2  50 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  oo 


DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "  "  "       Abridged  Ed 8vo,  i  50 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Advanced  Mechanical  Drawing. 8vo,  2  oo 

Jones's  Machine  Design : 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

MacLeod's  Descriptive  Geometry Small  8vo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i  50 

Industrial  Drawing.  (Thompson.) 8vo,  3  50 

Moyer's  Descriptive  Geometry 8vo,  2  oo 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.  (McMillan.) 8vo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

*  Titsworth's  Elements  of  Mechanical  Drawing Oblong  8vo, 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo, 


Drafting  Instruments  and  Operations i2mo, 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo, 

Plane  Problems  in  Elementary  Geometry i2mo, 


oo 

25 

50 

00 

25 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's    Kinematics    and    Power    of    Transmission.        (Hermann    and 

Klein.) 8vo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo.  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

10 


Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  i  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  oo 

ELECTRICITY  AND  PHYSICS. 

*  Abegg's  Theory  of  Electrolytic  Dissociation.     (Von  Ende.) i2mo,  i  25 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  i2mo,  i  oo 

Benjamin's  History  of  Electricity 8vo,  3  oo 

Voltaic  CelL 8vo,  3  oo 

Betts's  Lead  Refining  and  Electrolysis.     (In  Press.) 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).Svo,  3  oo 

*  Collins's  Manual  of  Wireless  Telegraphy 1 21110,  i  50 

Morocco,  2  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

*  Danneel's  Electrochemistry.     (Merriam.) i2mo,  i  25 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's  Theory  of  the  Lead  Accumulator  (Storage  Battery).    (Von  Ende.) 

i2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.). 8vo,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power 121110,  3  co 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  2  50 

Hanchett's  Alternating  Currents  Explained i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Hobart  and  Ellis's  High-speed  Dynamo  Electric  Machinery.     (In  Press.) 

Holman's  Precision  of  Measurements 8vo,  2  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and   Tests.  .  .  .Large  8vo,  75 
Karapetoff's  Experimental  Electrical  Engineering.     (In  Press.) 

Kinzbrunner's  Testing  of  Continuous-current  Machines.  ... 8vo,  2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,  3  oo 

Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo,  3  oo 

*  Lyons'?  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  8vo,  each,  6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,  2  50 

Norris's  Introduction  to  the  Study  of  Electrical  Engineering.     (In  Press.) 

*  Parshall  and  Hobart's  Electric  Machine  Design 410,  half  morocco,  12  50 

Reagan's  Locomotives:    Simple,  Compound,  and  Electric.      New  Edition. 

Large  i2mo,  3  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee— Kinzbrunner.).  .  .8vo,  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  50 

Thurston's  Stationary  Steam-engines 8vo,  50 

*  Tollman's  Elementary  Lessons  in  Heat 8vo,  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  06 

*  Sheep,  7  SO 

*  Dudley's  Military  Law  and  the  Procedure  of  Courts-martial  .  .  .   Largre  i2mo,  2  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Winthrop's  Abridgment  of  Military  Law 121110,  2  50 

11 


MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule 1 2mo,  2  50 

Holland's  Iron  Founder 1 2mo,  2  50 

The  Iron  Founder,"  Supplement \j 2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding 12010,  3  oo 

*  Claassen's  Beet-sugar  Manufacture.    (Hall  and  Rolfe.) 8vo,  3  oo 

*  Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Fitzgerald's  Boston  Machinist 12 mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo.  i  oo 

Herrick's  Denatured  or  Industrial  Alcohol ,8vo,    4   oo 

Holley  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments,  and  Varnishes. 

(In  Press.) 

Hopkins 's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i  50 

Maire's  Modern  Pigments  and  their  Vehicles.     (In  Press.) 

Matthews's  The  Textile  Fibres.     2d  Edition,  Rewritten 8vo,  4  oo 

Metcalf's  Steel.     A  Maunal  for  Steel-users i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops    ,8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i   50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Rice's  Concrete-block  Manufacture 8vo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish  .....  .8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion   8vo,  5  oo 

Ware's  Beet-sugar  Manufacture  and  Refining.     Vol.  I Small  8vo,  4  oo 

Vol.11 8vo,  5  oo 

Weaver's  Military  Explosives 8vo,  3  oo 

West's  American  Foundry  Practice 1 2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and  Steel     8vo,  4  oo 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo, 

Briggs's  Elements  of  Plane  Analytic  Geometry i2mo, 

Buchanan's  Plane  and  Spherical  Trigonometry.     (In  Press.) 

Compton's  Manual  of  Logarithmic  Computations i2mo, 

Da  vis's  Introduction  to  the  Logic  of  Algebra 8vo, 

*  Dickson's  College  Algebra Large  i2mo, 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2tno, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications  .....   8vo, 
Halsted's  Elements  of  Geometry 8vo, 

Elementary  Synthetic  Geometry 8vo, 

*  Rational  Geometry  .   i2mo, 

12 


5cr 
oo 


50 
50 
50 

25 

50 
75 
50 
50 


*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:  Vest-pocket  size. paper,         15 

100  copies  for     5  oo 

*  Mounted  on  heavy  cardboard,  8  X 10  inches,         25 

10  copies  for     2  oo 
Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .Small  8vo,     3  oo 

Elementary  Treatise  on  the  Integral  Calculus Small  8vo,     i   50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates 12010,     i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,     3  50 

Johnson's  Treatise  on  the  Integral  Calculus Small  8vo,     3  oo 

Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo,     i   50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics I2mo,     3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory. ).i2mo,     2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,     3  oo 

Trigonometry  and  Tables  published  separately Each,     2  oo 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,     i  oo 

Manning's  IrrationalNumbers  and  their  Representation  bySequences  and  Series 

i2mo,     i   25 
Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward Octavo,  each     i  oo 

No.  i.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
Ko.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
by  Robert  S.  Woodward.  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  o.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solution  of  Equations, 
by  Mansfield  Merriman.  No.  n.  Functions  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics. 8vo,    4  oo 

Merriman's  Method  of  Least  Squares 8vo,     2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,    3  oo 
Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,    2  50 

*  Veblen  and  Lennes's  Introduction  to  the  Real  Infinitesimal  Analysis  of  One 

Variable 8vo,    2  oo 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,     2  oo 

Trigonometry:   Analytical,  Plane,  and  Spherical 121110,     i  oo 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "                    "                  "         Abridged  Ed 8vo,  i   50 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  oo 

Carpenter's  Experimental  Engineering 8vo,  6  oo 

Heating  and  Ventilating  Buildings 8vo,  4  oo 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

13 


Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Gill's  Gas  and  Fu«l  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book. i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.    (Pope,  Haven,  and  Dean.) .  .  8vo,  4  oo 
MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

MacFar land's  Standard  Reduction  Factors  for  Gases 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  (O.)  Press- working  of  Metals 8vo,  3  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Thurston's   Treatise    on    Friction  and    Lost   Work   in    Machinery   and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  i2mo,  i  oo 

Tillson's  Complete  Automobile  Instructor i6mo,  i  50 

Morocco,  2  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.). 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

MATERIALS   OF   ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron : 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

14 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,    a  oo 

Elements  of  Analytical  Mechanics 8vo,    3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,    4  oo 

STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram . izmo,    i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) 12 mo,     i  50 

Creighton's  Steam-engine  and  other  Heat-motors         8vo,    500 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book i6mo,  mor.,    5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,     i  oo 

Goss's  Locomotive  Sparks 8vo      2  oo 

Locomotive  Performance 8vo,   5  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy .i2mo,    2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,    5  oo 

Heat  and  Heat-engines 8vo      5  oo 

Kent's  Steam  boiler  Economy 8vo,    4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,     i  50 

MacCord's  Slide-valves 8vo,    2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo      i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors    8vo,     i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,    5  oo 

Valve-gears  for  Steam-engines Svo,    2  50 

Peabody  and  Miller's  Steam-boilers 8vo,    4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,    2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) .i2mo,     i  25 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric.     New  Edition. 

Large  12 mo,    3  50 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,    2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,    2  50 

Snow's  Steam-boiler  Practice 8vo,    3  oo 

Spangler's  Valve-gears 8vo,    2  50 

Notes  on  Thermodynamics i2mo,    i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering  . 8vo,    3  oo 

Thomas's  Steam-turbines 8vo,    3  50 

Thurston's  Handy  Tables 8vo,    i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory. 8vo,    6  oo 

Part  II.     Design,  Construction,  and  Operation 8vo,    6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,    5  oo 

Stationary  Steam-engines 8vo,    2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,    i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation  8vo,    5  oo 
Wehrenfenning's  Analysis  and  Softening  of  Boiler  Feed-water  (Patterson)  8vo,     4  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,    5  oo 

Whitham's  Steam-engine  Design 8vo,    5  oo 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines. .  .8vo,    4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bovey  s  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making I2mo,  2  50 

15 


Church's  Mechanics  of  Engineering 8vo,  6  oo 

Notes  and  Examples  in  Mechanics 8vo,  2  oo 

Compton's  First  Lessons  in  Metal-working izmo, 

Compton  and  De  Groodt's  The  Speed  Lathe 'I2mo, 

Cromwell's  Treatise  on  Toothed  Gearing i2mo, 

Treatise  on  Belts  and  Pulleys i2mo, 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .12010, 

Dingey's  Machinery  Pattern  Making i2mo, 

Dredge's  Record  of  the   Transportation  Exhibits   Building  of  the   World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  4  oo 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist, i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Locomotive  Performance 8vo,  5  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  i  oo 

Hobart  and  Ellis 's  High-speed  Dynamo  Electric  Machinery.     (In  Press.) 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts .8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)- 8vo,  4  oo 
MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

*  Martin's  Text  Book  on  Mechanics,  Vol.  I,  Statics i2mo,  i   25 

*  Vol.  2,  Kinematics  and  Kinetics  .  .I2mo,  1  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Elements  of  Mechanics i2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

*  Parshall  and  Hobart's  Electric  Machine  Design 4to,  half  morocco,  12  50 

Reagan's  Locomotives :  Simple,  Compound,  and  Electric.     New  Edition. 

Large  i2mo,  3  So 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.  Vol.  1 8vo,  2  50 

Sanborn's  Mechanics :  Problems Large  i amo,  i  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines I2mo,  i  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design. 8vo,  3  oo 

Sorel' s  Carbureting  and  Combustion  of  Alcohol  Engines.  (Woodward  and 

Preston.) Large  8vo,  3  oo 

16 


Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo.  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  I2mo,  i  oo 

Tillson's  Complete  Automobile  Instructor , i6mo,  i   50 

Morocco,  2  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  So 

Weisbach's  Kinematics  and  Power  of  Transmission.   (Herrmann — Klein.). 8vo.  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.). 8vo.  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics 1 2mo,  i  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  I  oo 

MEDICAL. 

*  Bolduan's  Immune  Sera 12mo,  1  50 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.).    .  .  .Large  i2mo,  2  50 

Ehrlich's  Collected  Studies  on  Immunity.     (Bolduan.) 8vo,  6  oo 

*  Fischer's  Physiology  of  Alimentation Large  12mo,  cloth,  2  oo 

Hammarsten's  Text-book  on  Physiological  Chemistry.     (Mandel.) 8vo,  4  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.     (Lorenz.) i2mo,  oo 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.     (Fischer.) ....  12010,  25 

*  Pozzi-Escot's  The  Toxins  and  Venoms  and  their  Antibodies.     (Conn.).  i2mo,  oo 

Rostoski's  Sejum  Diagnosis.     (Bolduan.) i2mo,  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) 8vo,  50 

*  Satterlee's  Outlines  of  Human  Embryology i2mo,  25 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  50 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  oo 

Woodhull's  Notes  on  Military  Hygiene i6mo,  50 

*  Personal  Hygiene i2mo,  oo 

Wulling's  An  Elementary  Course  in  Inorganic  Pharmaceutical  and  Medical 

Chemistry i2mo,  2  oo 

METALLURGY. 

Betts's  Lead  Refining  by  Electrolysis.    (In  Press.) 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury; 

Vol.    I.     Silver , 8vo,  7  50 

Vol.  II.     Gold  and  Mercury 8vo,  7  50 

Goesel's  Minerals  and  Metals:     A  Reference  Book. , .  . . .  i6mo,  mor.  3  oo 

*  Iles's  Lead-smelting i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess. )i2mo,  3  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users 12010,  2  oo 

Miller's  Cyanide  Process I2mo,  i  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). , .  .  i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Smith's  Materials  of  Machines I2mo,  j  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

Part    II.     Iron  and  Steel 8VO)  3  5O 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

MINERALOGY. 
Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,    2  50 

Boyd's  Resources  of  Southwest  Virginia gvo,    3  oo 

17 


Boyd's  Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  i  s*> 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals Svo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  bvo,  i  oo 

Text-book  of  Mineralogy • 8vo,  4  oo 

Minerals  and  How  to  Study  Them I2mo,  I  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  oo 

Manual  of  Mineralogy  and  Petrography ismo  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects I2mo,  i  oo 

Eakle's  Mineral  Tables 8vo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo.  2  50 

Goesel's  Minerals  and  Metals :     A  Reference  Book ibmo,  mor.  3  oo 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall) 12 mo,  i  25 

Iddings's  Rock  Minerals 8vo,  5  oo 

Johannsen's  Key  for  the  Determination  of  Rock-forming  Minerals   in   Thin 
Sections.    (In  Press.) 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe.  I2mo,  60 
Merrill's  Non-metallic  Minerals.  Their  Occurrence  and  Uses 8vo,  4  oo 

Stones  for  Building  and  Decoration                      8vo,  500 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 

Tables  of  Minerals 8vo,  i  00 

*  Richards's  Synopsis  of  Mineral  Characters i2mo.  morocco,  i  25 

*  Ries's  Clays.  Their  Occurrence.  Properties,  and  Uses 8vo,  5  oo 

Rosenbusch's   Microscopical   Physiography   of   the   Rock-making  Minerals. 

(Iddings.) 8vo,  5  oo 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo,  2  oo 

MINING. 

Beard's  Mine  Gases  and  Explosions.     (In  Press.) 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virginia Pocket-book  form.  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  I  oo 

Eissler's  Modern  High  Explosives 8-73,  4  oo 

Goesel's  Minerals  and  Metals ;     A  Reference  Book i6mo,  mor.  3  oo 

Goodyear's  Coal-mines  of  the  Western  Coart  of  the  United  States i2mo,  2  50 

Ihlseng's  Manual  of  Mining.   8vo,  5  oo 

*  Bes's  Lead-smelting I2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Miller's  Cyanide  Process i2mo,  i  oo 

O'DriscolJ's  Notes  on  the  Treatment  of  Gold  Ores Svo,  2  oo 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) Svo,  4  oo 

Weaver's  Military  Explosives Svo,  3  oo 

Wilson's  Cyanide  Processes i2mo.  i  50 

Chlorination  Process limo,  i  50 

Hydraulic  and  Placer  Mining.     2d  edition,  rewritten i2mo,  2  50 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation I2mo,  i  25 

SANITARY  SCIENCE. 

Bastiore's  Sanitation  of  a  Country  House i2mo,  i  oo 

*  Outlines  ot  Practical  Sanitation I2mo,  I  25 

FolwelTs  Sewerage.     (Designing,  Construction,  and  Maintenance.) Svo,  3  oo 

Water-supply  Engineering Svo,  4  oo 

18 


Fowler's  Sewage  Works  Analyses .  .  , 121113,  2  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Sanitation  of  Public  Buildings 12mo,  1  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) 12010,  i  25 

*  Merriman's  Elements  of  Sanitary  Engineering 8vorf  2  oo 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

*  Price's  Handbook  on  Sanitation lamo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science 1 2mo,  i  oo 

Cost  of  Shelter i2mo,  i  oo 

Richards  and  Woodman's  Air.  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Si  wage  and  Bacterial  Purification  of  Sewage 8vo,  4  oo 

Disinfection  and  the  Preservation  of  Food 8vo,  400 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Air  Conditioning.    (In  Press.) 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woodhull's  Notes  on  Military  Hygiene iCmo,  i  50 

*  Personal  Hygiene I2mo,  i  oo 


MISCELLANEOUS. 

Association  of   State   and  National  Food  and  Dairy  Departments  (Interstate 
Pure  Food  Commission) : 

Tenth  Annual  Convention  Held  at  Hartford,  July  17-20,  1906.  ...8vo,     3  oo 
Eleventh    Annual    Convention,    Held  at  Jamestown   Tri-Centennial 

Exposition,  July  16-19,  1907.     (In  Press.) 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  Cvo,    i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,     4  oo 

Gannett's  Statistical  Abstract  of  the  World 24010,        75 

Gerhard's  The  Modern  Bath  and  Bath-houses.     (In  Press.) 

Haines's  American  Railway  Management I2mo,    2  30 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.. Small  8vo,    3  oo 

Rotherham's  Emphasized  New  Testament Large  8vo,    2  OQ 

Standage's  Decorative  Treatment  of  Wood,  Glass,  Metal,  etc.     (In  Press.) 

The  World's  Columbian  Exposition  of  1893 4to,    i  oo 

Winslow's  Elements  of  Applied  Microscopy I2mo,     i  50 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar 1 2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco.  5  oo 

Letteris's  Hebrew  Bible 8vo,  2  25 

19 


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